Optimal Consumption and Portfolio Selection with ... · The paper’s approach generalizes the Karatzas et al. [31], and Cox and Huang [5, 6] treatments2 of Merton’s [40] optimal

Journal of Economic Theory 89, 68�126 (1999)

Optimal Consumption and Portfolio Selectionwith Stochastic Differential Utility1

Mark Schroder

Department of Finance of the Eli Broad Graduate School of Management,Michigan State University, East Lansing, Michigan 48824

schrode7�pilot.msu.edu

and

Costis Skiadas

Department of Finance of the J. L. Kellogg Graduate School of Management,Northwestern University, Evanston, Illinois 60208

c-skiadas�nwu.edu

Received November 21, 1997; revised May 19, 1999

We develop the utility gradient (or martingale) approach for computing portfolioand consumption plans that maximize stochastic differential utility (SDU), acontinuous-time version of recursive utility due to D. Duffie and L. Epstein (1992,Econometrica 60, 353�394). We characterize the first-order conditions of optimalityas a system of forward�backward SDEs, which, in the Markovian case, reduces toa system of PDEs and forward only SDEs that is amenable to numerical computa-tion. Another contribution is a proof of existence, uniqueness, and basic propertiesfor a parametric class of homothetic SDUs that can be thought of as a continuous-time version of the CES Kreps�Porteus utilities studied by L. Epstein and A. Zin(1989, Econometrica 57, 937�969). For this class, we derive closed-form solutions interms of a single backward SDE (without imposing a Markovian structure). Weconclude with several tractable concrete examples involving the type of `àffine''state price dynamics that are familiar from the term structure literature. Journal ofEconomic Literature Classification Numbers: G11, E21, D91, D81, C61. � 1999

Academic Press

Article ID jeth.1999.2558, available online at http:��www.idealibrary.com on

680022-0531�99 �30.00Copyright � 1999 by Academic PressAll rights of reproduction in any form reserved.

1 We are grateful for helpful discussions with John Campbell, George Constantinides, KentDaniel, Darrell Duffie, Bernard Dumas, Janice Eberly, Mark Fisher, John Heaton, RaviJagannathan, Erzo Luttmer, Lars Nielsen, Ken Singleton, Raman Uppal, Luis Viceira, andTan Wang. We are responsible for any errors. Results omitted in the published version canbe downloaded from http:��www.kellogg.nwu.edu�faculty�skiadas�research�research.htm.

1. INTRODUCTION

This paper develops the utility gradient, or martingale, approach forsolving optimal consumption-portfolio selection problems in continuous-time complete markets with Brownian information and stochastic differentialutility (SDU). SDU was introduced by Duffie and Epstein [12] as acontinuous time limit of the type of recursive utility studied by Kreps andPorteus [35], Epstein and Zin [24], and many others.

The importance and structure of preferences that are not necessarilytemporally additive have been studied extensively in a literature surveyedby Epstein [23]. It is, for example, well known that utility additivity withrespect to time and states of nature is overly restrictive in expressingreasonable notions of risk aversion in temporal settings. (The introductionof Duffie and Epstein [12] gives a suggestive example.) As Epstein [23]explains in his survey, under additive temporal preferences notions ofintertemporal substitution and risk aversion are inflexibly linked to eachother. SDU, while nesting the time-additive case, is more flexible in thisregard, capturing the notion that one's present sense of well-being candepend on one's expected future utility levels in a not necessarily risk-neutral manner. This effect is also related to attitudes toward the timing ofresolution of uncertainty, as discussed, among others, by Kreps andPorteus [35], Chew and Epstein [3, 4], and Skiadas [47], the lastreference covering the case of SDU.

The paper's approach generalizes the Karatzas et al. [31], and Cox andHuang [5, 6] treatments2 of Merton's [40] optimal portfolio selectionproblem with additive utilities (textbook accounts of which are given byMerton [41], Duffie [11], and Karatzas and Shreve [33]). The basic ideais to utilize market completeness to separate the computation of an optimalconsumption plan and that of a corresponding trading strategy. Theoptimal consumption is obtained by solving the first-order conditions,essentially stating that the agent's marginal utility process at the optimumis proportional to an Arrow�Debreu state price density process. A corre-sponding financing trading strategy can then be constructed using Merton'sstandard replication arguments (developed in the context of the Black�Scholes theory of option pricing). Duffie and Skiadas [20] showed thatthis approach remains valid quite generally, even with non-additivepreferences, provided that a marginal utility process is well defined asthe Riesz representation of the (infinite-dimensional) utility gradient at theoptimum. Moreover, they obtained closed-form expressions for marginalutilities in this sense for various types of temporally dependent preferencesused in practice, including SDU. Even earlier, Duffie and Epstein [12]

69OPTIMAL PORTFOLIO SELECTION

2 Closely related are the works of Pliska [44] and Foldes [26].

used dynamic programming methods to compute the form of Arrow�Debreu prices under SDU in a more special Markovian setting underBrownian information. The Duffie�Epstein�Skiadas papers, however, areoriented toward obtaining equilibrium pricing formulas and do not in factsolve the first-order conditions for the optimal consumption plan.

With additive utilities, the solution of the first-order conditions amountsto a straightforward inversion of the marginal utility at each state�timeseparately. With SDU, the solution of the first-order conditions involvesthe solution of a system of so-called forward�backward stochastic differentialequations (FBSDEs), a fixed-point problem involving the complete infor-mation filtration at once. The mathematical theory of FBSDEs is fairlyrecent, and a satisfactorily general existence theory is missing at themoment. Ma et al. [39] have developed a useful scheme for solvingFBSDEs in Markovian settings with Brownian information via thesolution of quasilinear partial differential equations (PDEs). A firstcontribution of this paper is to import the methodology of Ma et al. to theproblem of optimal portfolio selection in a Markovian setting withBrownian information. The result is a remarkably simple solution proce-dure that complements the more traditional dynamic programmingapproach and in fact results in a numerically more tractable PDE formthan that associated with the Bellman equation. A similar approach applieswith other temporally dependent utility forms whose gradients are com-puted by Duffie and Skiadas [20]. In order to keep this paper focused andof manageable size, we will restrict our analysis to the SDU case. Schroderand Skiadas [46] present extensions that include habit formation.

In addition to providing a general solution method, we will apply thisapproach to a parametric class of homothetic SDU that was introducedby Duffie and Epstein [12, 13] as a continuous-time limit of the CESKreps�Porteus specification studied by Epstein and Zin [24] and by Weil[51]. Without Markovian assumptions, we will show that the first-orderconditions simplify significantly in this case, resulting in closed-formexpressions in terms of the solution to a single backward SDE. The latterbecomes trivial under a deterministic investment opportunity set, while inmore general Markovian settings the solution of this backward SDEreduces to the solution of a numerically tractable PDE. Analytic solutionswill be derived for examples involving a stochastic investment opportunityset. A difficulty with this parametric SDU class is that both the utilityspecification and the associated first-order conditions involve backwardSDEs that violate the usual Lipschitz growth assumptions of availableexistence results by Pardoux and Peng [43] and Duffie and Epstein [12].Instead, we will prove existence, uniqueness, and basic properties from firstprinciples using monotonicity arguments. Another difficulty is that most of

70 SCHRODER AND SKIADAS

the technical restrictions required by the results of Ma et al. [39] andDuffie and Skiadas [20] are also violated in this context. Again, we willprovide the requisite special arguments that prove optimality of theproposed solutions.

For some of the utility parameter ranges we consider, Svensson [49]and Obstfeld [42] have provided heuristic derivations of the solutions forthe special case of a constant (deterministic) investment opportunity set.They have not addressed, however, the issues of utility function existenceand optimality verification. Fisher and Gilles [25] have also worked, con-currently with this paper3, on a parametric homothetic case in an infinitehorizon setting with a stochastic investment opportunity set. While theirexamples can be viewed as infinite horizon limits of examples we analyzein this paper, the two papers have different objectives. Fisher and Gillesmanipulate the utility forms and first-order conditions heuristically andproceed quickly to numerical solutions of PDEs. In contrast, the presentpaper contains no numerical examples, but instead concentrates on thetheoretical development, which, as noted above, includes the requisitebackward SDE theory and optimality verification arguments.

Discrete time versions of some of our homothetic examples have beenanalyzed by Giovannini and Weil [29] and Campbell and Viceira [2]. Theformer concentrate on special cases resulting in myopic portfolios or con-sumption plans, and the latter use an approximate log-linearization of thebudget constraint. Some special cases involving additive utility and specificparameterizations of price dynamics are closely related to those of Kim andOmberg [34], Liu [37], and Wachter [50] (the latter two were writtenconcurrent with and independent of the present paper.) Finally, this paperis related to several other papers that discuss the use of FBSDEs indifferent contexts of finance theory, including Cvitanic� [7]; Cvitanic� andMa [8]; Duffie et al. [14, 17, 18, 19]; and El Karoui et al. [22]. The lastreference also discusses a dual characterization of SDU, introduced byGeoffard [28] (for the deterministic case) and used by Dumas et al. [21]in their study of efficient allocations with SDU.

The remainder of this paper is organized into six sections and threeappendixes. Section 2 gives an abstract formulation of the problem,


3 Fisher and Gilles originally used an early version of our paper that included the generalsolution method, but not the current Theorems 3 and 4. They independently derived heuristicversions of these results for the infinite horizon case. Their discussion offers useful intuitionand complements ours. They did not address, however, the questions of existence of the utilityfunction being maximized and of the stochastic process used to describe the candidate optimalsolution. Neither did they prove that the proposed solution is in fact optimal. More recentversions of their paper make full use of our results as reported here.

presenting the first-order conditions of optimality in terms of Arrow�Debreuprices and the Riesz representation of the utility gradient. Section 3 showshow the general analysis applies with SDU and outlines a computationalapproach in Markovian settings. Section 4 introduces a parametric class ofhomothetic SDU, develops their basic properties, and presents optimalconsumption and portfolio rules in closed form for several special cases.The general solution method for this SDU class is discussed in Section 5,followed in Section 6 by several examples with a stochastic investmentopportunity set. Section 7 concludes with an outline of how to incorporatea bequest function in the analysis and a parametric example. Theappendixes contain proofs, and they develop some requisite mathematicaltheory.

2. ABSTRACT FORMULATION

We begin with a probability space (0, F, P) supporting a n-dimensionalstandard Brownian motion, B, over the finite time horizon [0, T]. Allstochastic processes introduced in this paper will be assumed progressivelymeasurable with respect to the augmented filtration [Ft : t # [0, T]]generated by B. We also assume that F=FT . The conditional expectationoperator E[ } | Ft] will be abbreviated to Et throughout.

We let D denote the Hilbert space containing any progressivelymeasurable process of the form x : 0_[0, T] � R satisfying E(�T

0 x2t dt)

<�. The inner product of D is defined by (x, y) =E(�T0 xt yt dt). As

usual, we identify any x, y # D such that (x& y, x& y) =0. The space ofall processes is partially ordered by letting x�(>) y denote the condition:x(|, t)�(>) y(|, t) for almost every (|, t). The positive cone ofD is D+=[x # D : x�0], and the strictly positive cone is D++=[x # D : x>0]. For mathematical background material we refer to Karatzasand Shreve [32].

2.1. Optimality in an Arrow�Debreu Market

We take as primitive a consumption space C, assumed throughout tosatisfy:

C1. C is a cone that is a subset of D+ and has the property that forevery bounded h # D+ and c # C, c+h # C.

For example, we can take C=D+, but in our main application we willneed to impose additional integrability conditions on elements of C. Aconsumption process, c, is any element of C, with ct representing a time tconsumption rate in terms of a single numeraire good. For simplicity of


exposition, we do not allow terminal consumption, although our analysisextends in a straightforward manner to include it, as outlined in theconcluding section.

We will consider an agent with initial wealth w>0 that maximizes autility function V0 : C � R by trading in a complete securities market (thatis, a market in which every consumption plan can be financed givensufficient initial wealth).

Given c # C, we let F(c) denote the set of feasible directions at c, that is,the set of any h # D such that c+:h # C for all sufficiently small positive :.The Gateaux derivative of V0 at c is defined by

{V0(c; h)=lim: a 0

V0(c+:h)&V0(c):

, h # F(c).

This derivative is closely related to a function m : C � D, which we takeas given throughout the paper, and, for SDU, will be given in closed form.We will use m through one of two conditions. The first condition, satisfiedunder general assumptions on the utility form (see Duffie and Skiadas[20]), but not always, is

C2. For every c # C, {V0(c; h)=(m(c), h) for all h # F(c).

In particular, C2 implies that, for every c, the Gateaux derivative of V0

at c exists and is linear. Under these conditions, {V0(c, } ) is called theutility gradient of V0 at c, with Riesz representation m.

In some applications, condition C2 is unnecessarily strong for the pur-pose of confirming optimality of a given consumption plan, and we willinstead use the following inequalities, where c # C is some candidateoptimal consumption plan:

C2$. For all c # C, V0(c)�V0(c)+(m(c), c&c) .

In complete markets, the determination of an optimal consumption plandepends only on preferences, endowments, and Arrow�Debreu state prices,while the corresponding financing strategy can be derived using standardreplication arguments. An Arrow�Debreu state price density is any strictlypositive process, ? # D++, such that capital plus dividend gains from tradedeflated by ? form a martingale. Bypassing, for now, the familiar derivationof state price dynamics from price dynamics, we take as primitive a(normalized) state price density process, ?, with dynamics

&d?t

?t=rt dt+'$t dBt , ?0=1,


where �T0 |rt |+'$t't dt<� a.s. The process r is the short-rate process, and

' is the market-price-of-risk process.4

The consumption process c is defined to be optimal if

c # arg max[V0(c) : (?, c)�w, c # C].

2.2. First-Order Conditions

Given C2, the first-order conditions for optimality of c # C take thefamiliar form

mt(c)=*?t on [ct>0] for a.e. t # [0, T],

mt(c)�*?t on [ct=0] for a.e. t # [0, T], (1)

(?, c)�w, *(w&(?, c) )=0, * # R+.

The specification `òn'' some event in (1), and throughout the paper, shouldbe interpreted in the almost sure sense.

Proposition 1. (a) Suppose that conditions C1 and C2 hold. Then (1)holds for every optimal c # C. (b) Suppose that conditions C1 and C2$ hold,for some c # C. If (1) is satisfied with c=c, then c is optimal. (c) If V0 is con-cave and continuous and C is convex and closed, then an optimal consumptionprocess exists.

Proof. (a) Consider the following first-order conditions for optimalityof c # C:

{V0(c; h)&*(?, h)�0, h # F(c),

(?, c)�w, *(w&(?, c) )=0, * # R+.

By the generalized Kuhn�Tucker theorem, these conditions are necessaryfor optimality of c. For example, Theorem 1 of Section 9.4 of Luenberger[38] can be used, with only a straightforward modification of the proof toaccount for the fact that the underlying space is a positive cone, and nota linear space. Given C1 and the utility gradient Riesz representation, m,the above conditions imply that for all bounded h # D+, (m(c)&*?, h)�0, and therefore mt(c)�*?t a.s. for almost every t. Since we also have


4 An equivalent formulation results if, instead of state prices, we consider an equivalentmartingale measure, in the sense of Harrison and Kreps [30], relative to the short-rateprocess r. This, for example, is the approach taken by Cox and Huang [5]. For the purposesof this paper, the formulation in terms of state prices is notationally more parsimonious.A textbook exposition of the relationship between state prices and equivalent martingalemeasures can be found in Duffie [11].

(m(c)&*?, :c) �0 for :> &1, mt(c)=*?t on [ct>0] for almost every t.This proves part (a).

(b) If (1) holds with c=c, then (m(c), c&c) �0 for any budgetfeasible c # C, and the result follows from the gradient inequality ofcondition C2$.

(c) If V0 is concave and norm-continuous, it is weakly upper-semi-continuous. By Alaoglu's theorem, and since for convex sets strong closureis the same as weak closure, the budget feasible set is weakly compact. Themaximum is therefore achieved. K

2.3. Securities Market

The above discussion is all in the setting of an Arrow�Debreu market.The implementation of the optimal consumption plan by trading in asecurities market is well understood. In order to fix notation in discussingapplications later on, we now outline a simple securities market thatimplements the above Arrow�Debreu market. More general formulationsand details can be found in the expositions of Duffie [11] and Karatzasand Shreve [33].

In addition to short-term default-free borrowing and lending at a rategiven by the process r, there are n risky securities (one for each componentof the Brownian motion B). The risky asset instantaneous excess returns(relative to r) are represented by the n-dimensional Ito process Rt=[R1

t , ..., Rnt ]$, with decomposition

dRt=+Rt dt+_R

t dBt ,

where +R and _R are progressively measurable processes valued in Rn andRn_n, respectively, and satisfy �T

0 |+Rt |+_R

t (_Rt )$ dt<� a.s. We assume that

_Rt is invertible almost everywhere, and 't=(_R

t )&1 +Rt .

A trading strategy is any progressively measurable process, �, valued inRn, such that � t

0 |�$s +Rs |+�$s_R

s (_Rs )$ �s ds<� a.s. for all t<T. We inter-

pret � it as the time t proportion of wealth invested in security i # [1, ..., n],

the remaining wealth being invested at the short rate r. Given any initialwealth w, consumption plan c, and trading strategy �, the correspondingwealth process W w, c, � is defined by the budget equation:

dW w, c, �t =W w, c, �

t (�$t dRt+rt dt)&ct dt, t<T, W w, c, �0 =w.

We say that � finances c given initial wealth w if W w, c, �t �0 a.s. for all t<T.

For every consumption plan c, we define the process

Wt(c)=1?t

Et _|T

t?scs ds& .


Recalling the normalization ?0=1, we have W0(c)=(?, c) . It can beshown (see, for example, Karatzas and Shreve [33]) that for every con-sumption plan c, there exists a trading strategy � that finances c giveninitial wealth w=(?, c) , such that W(c)=W w, c, �. This trading strategycan be calculated in terms of the wealth dynamics dWt(c)�Wt(c)=+W(c)

t dt+_W(c)t dBt , by matching the diffusion term with that of the budget

equation, to obtain �=(_R$t )&1 _W(c)$

t . Conversely, if c can be financedgiven some initial wealth w, then (?, c)�w. These familiar results implythat the problem of determining an optimal trading strategy given initialwealth w reduces to the optimal consumption problem of Section 2.1.

3. SOLUTION METHOD WITH STOCHASTIC DIFFERENTIALUTILITY

The general approach of the last section is further developed in thissection for the special case of stochastic differential utility.

3.1. Stochastic Differential UtilityStochastic differential utility is defined in terms of a function f : [0, T]_

R+_R � R, called the intertemporal aggregator. We refer to the threearguments of f as the time, consumption, and utility argument, respectively,and we write fc and fv for the partial derivatives of f with respect to theconsumption and utility arguments, respectively. To every c # C we assigna utility process, V(c), that satisfies

Vt(c)=Et _|T

tf (s, cs , Vs(c)) ds& , t # [0, T]. (2)

The following condition will be assumed throughout:

C3. There exists V�D such that, for every c # C, a unique V(c) # V

satisfies (2). The utility function V0 : C � R is defined by the initial valuesof V(c) # V, c # C, and the function m : C � D (appearing in C2 or C2$) isgiven by

mt(c)=exp \|t

0fv(s, cs , Vs(c)) ds+ fc(t, ct , Vt(c)). (3)

The combined results of Duffie and Epstein [12] and Duffie and Skiadas[20] show that conditions C2 and C3 (with C=D+ and V=D) hold iff satisfies certain continuity-Lipschitz-growth type conditions.5 Moreover,


5 More specifically, it is sufficient that f be continuously differentiable in its consumptionand utility arguments, and for some constant K, | f (t, c, 0)|�K(1+c), | fc(t, c, v)|�K(1+c+|v| ), and | fv(t, c, v)|�K, for all t # [0, T], c # R+, and v # R.

under the same restrictions on f, V0 is continuous, V0 is increasing if f isincreasing in its consumption argument, and V0 is concave if f is concavejointly in its consumption and utility arguments.

Unfortunately, the conditions on f imposed by the Duffie�Epstein�Skiadas results are violated in the parametric cases discussed later in thispaper. We will therefore present the general solution method by directlyassuming the validity of conditions C1, C2 or C2$, and C3, as well as thefollowing restriction on f whose purpose is to simplify the first-orderconditions for optimality:

C4. For all t # [0, T], c # R+, and v # R, fc(t, c, v)>0 andlimx � � fc(t, x, v)=0.

In applications, the above conditions will be verified on a case-by-casebasis.

For the case of a concave intertemporal aggregator, the following resultprovides a direct justification for the expression (3) and an easy way ofconfirming C2$, and hence the sufficiency of the first-order conditions foroptimality.

Lemma 1. Suppose that condition C3 is satisfied, f (t, } , } ) is concave( jointly in consumption and utility) for all t # [0, T], and for some c # C,

E _|T

0(max[ fv(t, ct , Vt(c)), 0])2 dt&<�.

Then condition C2$ is satisfied (with m defined in Eq. (3)).

Lemma 1 implies in particular that V0 is concave if f is concave jointlyin its consumption and utility arguments, provided that the assumedintegrability condition holds for any c # C. This conclusion generalizesProposition 5 of Duffie and Epstein [12] by allowing fv to be unbounded.Later we will encounter applications in which V0 is concave, but f is notjointly concave in consumption and utility.

3.2. First Order Conditions

Let the function I : [0, T]_R2 � R+ be defined by

I(t, x, v)=0, if ex� fc(t, 0, v);

fc(t, I(t, x, v), v)=ex, if ex< fc(t, 0, v).


Assuming that V0 is strictly increasing (a consequence of C2 and C4), thefirst-order conditions under SDU are equivalent to the system

Xt=log(*)&|t

0 \fv(s, I(s, Xs , Vs), Vs)+rs+12 '$s's+ ds

&|t

0'$s dBs , (4a)

Vt=Et _|T

tf (s, I(s, Xs , Vs), Vs) ds& , (4b)

E _|T

0?t I(t, Xt , Vt) dt&=w, * # R++. (4c)

Proposition 2. Given C1, C2, C3, and C4, every optimal consumptionprocess c # C takes the form ct=I(t, Xt , Vt), t # [0, T], where (X, V, *) #D_V_R++ is a solution to (4). Conversely, if conditions C1, C2$, C3, andC4 hold for a given c # C of this form, then c is optimal.

Proof. Defining Xt(*, c)=log(*?t)&�t0 fv(s, cs , Vs(c)) ds, the first-order

conditions, under our assumptions, can be written as ct=I(t, Xt(*, c), Vt(c))and (?, c) =w. The result is an easy consequence of this observation andProposition 1. K

The system of equations (4a), (4b) is a forward�backward stochasticdifferential equation (FBSDE) system, X being the forward component,and V being the backward component. General existence-uniquenessresults for this type of equations that do not rely on additional Markovianstructure and PDE techniques are lacking. Antonelli [1] presentssome related results under conditions that are likely to be violated inthis context. Antonelli's examples illustrate what can go wrong if hisassumptions are relaxed. In our context, Proposition 1(c) shows that asolution to (4) is guaranteed if V0 is continuous and concave, and C isclosed and convex.

3.3. Computational Approach

Computationally, system (4) can be approached by the methodology ofMa et al. [39], as we now show. A Markovian structure is required for thispurpose. We therefore assume that the following condition holdsthroughout this subsection:


C5. (a) The functions +Y : [0, T]_Rm � Rm, and _Y : [0, T]_Rm � Rn are such that the following SDE has a unique (strong) solution,Y, valued in Rm:

Yt=Y0+|t

0+Y (s, Ys) ds+|

t

0_Y(s, Ys) dBs .

(b) For some (measurable) functions r : [0, T]_Rm � R and' : [0, T]_Rm � Rn, rt=r(t, Yt) and 't='(t, Yt).

In part (b) we use r and ' to represent both the stochastic processes andthe corresponding functions of the underlying Markov state. The meaningwill always be clear from the context. Restrictions on +Y and _Y thatguarantee C5(a) are well known and can be found, for example, in Karatzasand Shreve [32]. In the case in which r and ' are deterministic processes,the process Y should be taken to be the empty function.

The key idea is that in a solution of (4), Vt should be a function of(Xt , Yt), for every t. So let us hypothesize the existence of a functiong : [0, T]_Rm+1 � R such that Vt= g(t, Xt , Yt) for all t. The dynamics ofX become

Xt=log(*)+|t

0\(s, Xs , Ys , g(s, Xs , Ys)) ds&|

t

0'(s, Ys)$ dBs , (5a)

where the function \ is defined by

\(t, x, y, v)=&fv(t, I(t, x, v), v)&r(t, y)& 12'(t, y)$ '(t, y). (5b)

Applying Ito's lemma, we are led naturally to the following quasilinearPDE for g (where the arguments of g and its derivatives are omitted):

& f (t, I(t, x, g), g)= gx \(t, x, y, g)+ gy+Y (t, y)+ gt& g$xy_Y (t, y) '(t, y)

+ 12 gxx'(t, y)$ '(t, y)

(6)+ 1

2 tr(gyy_Y (t, y) _Y (t, y)$),

g(T, } , } )=0.

Notice that the above PDE does not involve the Lagrange multiplier *.

Proposition 3. Suppose that C5 holds, g solves PDE (6), and

E _\|T

0&gx(t, Xt , Yt) '$t(t, Yt)& gy(t, Xt , Yt) _Y (t, Yt)&2 dt+

1�2

&<�.

(7)


For any *>0, if X solves SDE (5), and Vt= g(t, Xt , Yt), then (X, V ) solvesthe FBSDE system (4a), (4b).

Proof. The result is an immediate consequence of Ito's lemma,condition (7) ensuring that the local martingale part of the expansion is amartingale. K

A numerical implementation of this approach to solving the FBSDEsystem (4a), (4b) can be based on the procedure of Douglas et al. [10].

Propositions 2 and 3 imply that if we can find a function g that solves(6) and satisfies (7), then the agent's problem reduces to the simple task offinding a value for * that makes the budget constraint tight. Given theMarkovian specification of optimal consumption, it is a standard exerciseto find a corresponding financing trading strategy in short-term borrowingor lending and m+1 funds, each one of which is chosen to be instan-taneously perfectly correlated with X or a component of Y.

The results of Ma et al. [39] show that the following restrictions are suf-ficient for the existence of a solution to (6) that satisfies (7) (in particularimplying that gx and gy are bounded):

(a) There exists =>0 such that, for all (t, y) # [0, T]_Rm, thesmallest eigenvalue of ['(t, y), _Y (t, y)$]$ ['(t, y), _Y (t, y)$] is at least =.

(b) The functions '(t, y), +Y (t, y), _Y (t, y), and f (t, I(t, x, 0), 0) areall bounded, and |\(t, x, y, v)|�,( |v| ) for all x, y, v and t, for somefunction ,.

In the case of deterministic r and ', the conditions simplify further to (a)' is bounded away from zero, and (b) the function f (t, I(t, x, 0), 0) isbounded, and |\(t, x, v)|�,( |v| ) for all x, v, and t, for some function ,. Allof these conditions are far from necessary for Proposition 3 to apply,however. In fact, in the remainder of this paper we will discuss exclusivelyproblems that violate the above conditions.

4. A CLASS OF HOMOTHETIC SDU

This section analyzes a parametric homothetic SDU specification, forwhich the solution method of the last section simplifies significantly. Themain results are existence and basic properties of the utility function andthe sufficiency of the first-order conditions for optimality. The section con-cludes with expressions for optimal consumption plans and portfolios insome simple special cases. The general solution of the agent's problem forthis SDU class is the topic of the following section.


4.1. Utility Specification

We begin by restricting the space of consumption processes:6

C={c # D++ : E \|T

0c l

t dt+<� for all l # R= . (8a)

The intertemporal aggregator we consider takes one of the forms

f (c, v)={(1+:)((c#�#) |v| :�(1+:)&;v),(1+:v)[log(c)&(;�:) log(1+:v)],

if #{0,if #=0,

(8b)

with parameter restrictions:7

;�0 and {:>&1, and #<min[1, (1+:)&1],:�;,

if #{0,if #=0.

(8c)

We have omitted the time argument of f, since f is time-independent in thisformulation. The consumption argument, c, is restricted to be strictly positive,and for #=0, the utility argument, v, is restricted to be greater than &1�:. For:=#=0 we interpret (8b) by taking a limit, that is, f (c, v)=log(c)&;v.

Ordinally equivalent utility processes are defined by

V 1�(1+:)t , if #>0;

V� t={&|Vt|1�(1+:), if #<0; (9)

:&1 log(1+:Vt), if #=0.

The backward SDE satisfied by this version of the utility takes the moreintuitive form (omitting the argument of V� =V� (c))

V� t={Et _|

T

te&;(s&t)\(c#

s �#) ds+(:�2) V� &1s d[V� ]s+& ,

Et _|T

te&;(s&t)\log(cs) ds+(:�2) d[V� ]s+& ,

if #{0,

if #=0,(10)

where [V� ] denotes the quadratic variation of V� .


6 For simplicity, the integrability restrictions we impose are stronger than necessary for ourproofs to go through, but have the advantage that they are independent of utility parameters.

7 For #{0, the restriction :> &1 is assumed to keep utility finite near T. This issue canbe easily finessed by introducing a non-zero bequest function (with arbitrarily small weight),as outlined in Section 7, where we present an interesting example with :=&1. For :< &1the utility function to be maximized must be defined as U(c)=&V0(c). Concavity andmonotonicity of U (given the second inequality of (8c)) can then be proved as in Theorem 1.

The parameter : clearly has no impact on preferences over the set ofdeterministic consumption paths. It is a measure of comparative risk aver-sion in a sense defined by Duffie and Epstein [12], as well as a measureof preferences for the timing of resolution of uncertainty in a sense definedby Skiadas [47] (for a related application, see Duffie et al. [19]). If #>0,then V� t>0 for all t<T, and a negative : penalizes variability of the utilityprocess. Therefore, if #>0, risk aversion increases as the value of :decreases, a negative : indicates preferences for early resolution, and apositive : indicates preferences for late resolution. The same conclusionsare valid if #=0, although V� need not be uniformly signed in this case.Finally, if #<0, the role of the sign of : is reversed, because in this caseV� t<0 for all t<T. Therefore, with #<0, risk aversion increases with :,and a positive (negative) : corresponds to preferences for early (late)resolution of uncertainty.

4.2. Existence and Basic Properties

The above SDU specification is ordinally equivalent to a parametricclass first studied by Duffie and Epstein [12, 13], who also explain in whatsense the functional form for #=0 is a limiting case (under appropriatenormalization) of the case for #{0. The above SDU can also be obtainedas a continuous-time limit of the CES Kreps�Porteus utility specificationused by Epstein and Zin [24]. Nevertheless, a general proof of existenceand basic properties has been lacking. The aggregator, f, defined in (8)violates the usual Lipschitz and growth conditions used by Duffie andEpstein [12] and Pardoux and Peng [43] to prove existence and unique-ness of the backward SDE underlying the SDU definition. Duffie and Lions[16] give a PDE characterization of the utility process (for the infinitehorizon case), but their Markovian restrictions on consumption plans arenot appropriate in the current setting, and their parameter restrictions ruleout interesting parameter ranges satisfying (8c). Instead, we will base ourproof of existence and basic properties on our probabilistic results on back-ward SDEs of Appendix A.

To state the main existence result, we introduce the space

D0=[V # D : E[ess supt

|Vt|l ]<�, for every l>0],

and its strictly positive cone D++0 =[V # D0 : V>0 almost everywhere].

Throughout our discussion of the utility specification (8), we will assumethat the utility process set V is given as

V={[V : #V # D++0 ],

[V : 1+:V # D++0 and (1+:V )&1 # D0],

if #{0;if #=0.


Theorem 1. Suppose that C and f are specified by (8). Then recursion(2) is satisfied by a unique V(c) # V. Moreover, V0 is strictly concave,increasing, and homothetic.8

Other interesting properties of V0 involve comparative risk aversion andpreferences for the timing of resolution of uncertainty, as briefly discussedabove. The respective formal treatments of Duffie and Epstein [12,Proposition 6] and Skiadas [47, Appendix A] are based on Lipschitz con-ditions that are violated here. Given our results in Appendix A, however,adapting their arguments to the present setting becomes a tractable exercisethat we leave to the interested reader.

A relevant utility property that Theorem 1 does not cover is the existenceand form of the utility gradient. (Once again, the utility gradient computa-tions of Duffie and Skiadas [20] do not apply in this context.) Due tointegrability issues, condition C2 may not be satisfied for all parametervalues specified in (8c). However, since the unique optimal consumptionplan will later be specified in closed form, all that we need is the verifica-tion of condition C2$ at the optimum, which is accommodated by thefollowing result.

Lemma 2. Suppose that C and f are specified by (8). For condition C2$to hold with m given by Eq. (3), it is sufficient that either :#=0, or :#{0and E[exp(3 �T&=

0 | fv(ct , Vt(c))| ) dt]<� for every = # (0, T ).

The integrability condition of Lemma 2 is required only when both :{0and #{0, and even then it can be weakened, as indicated in the proof ofLemma 2. Nevertheless, the above form of the lemma will suffice foroptimality verification in our applications below.

4.3. Optimal Consumption and Portfolios for Some Special Cases

Significant simplifications to the solution of the agent's problem result ifone or more of the following conditions are satisfied:

I. Time-additive utility: :=0.

II. Logarithmic SDU: #=0.

III. Deterministic investment opportunity set: r and ' are deterministic.

For easy reference, we summarize these simplifications in the followingtheorem, making use of two auxiliary processes,

kt={;[;&:(1&e&;(T&t))]&1,(1&#(1+:))&1,

if #=0;if #{0

(11)


8 V0 is homothetic if, for all k>0, V0(c)�V0(c$) � V0(kc)�V0(kc$).

and

qt=;

1&#&

#1&# \rt+

kt

2't } 't+ .

The form of the solution below under a deterministic investment oppor-tunity set has been previously derived by Svensson [49] and Obstfeld [42](who, however, did not prove optimality or that the utility is well defined.)Here, and in the following section, we will make the assumption that r and' are bounded in order to avoid long technical discussions relating tointegrability issues. Depending on parameters, several of our proofs applywith weaker restrictions on r and '.

Theorem 2. Suppose that C and f are specified by (8), and r and ' arebounded.

(a) Suppose that at least one of conditions I and III holds. Then theoptimal consumption plan follows the dynamics

dct

ct=(rt&qt+kt't } 't) dt+kt '$t dBt .

(b) Suppose that at least one of conditions II and III holds. Then theoptimal consumption to wealth ratio is

ct

Wt(c)=_|

T

texp \&|

s

tq{ d{+ ds&

&1

.

(c) Suppose that either condition III holds, or conditions I and II hold(that is, :=#=0), or all three conditions hold. Then the optimal tradingstrategy is

�t=kt(_Rt _R$

t )&1 +Rt .

Under the assumptions of part (c) of the theorem, the optimal portfoliois instantaneously mean�variance efficient, even without additivity. But,compared to the Merton solution (:=0), less is invested in the risky fundif there are preferences for early resolution of uncertainty (#:<0, or #=0and :<0), and more is invested in the risky fund if there are preferencesfor late resolution (#:>0, or #=0 and :>0). An interesting observationis that if #=0, and r and ' are deterministic and constant over time, then,where Merton's solution involves constant over time portfolio weights, for:{0, the optimal positions change deterministically over time, approachingthe Merton solution toward the end of the planning horizon, T.


An example with non-additive SDU in which the optimal portfolio isinstantaneously mean�variance efficient under any state price dynamics isoutlined in Section 7.

5. GENERAL SOLUTION METHOD FOR HOMOTHETICSDU CLASS

This section presents a general solution method for the SDU specifica-tion of Theorem 1 and a stochastic investment opportunity set.

5.1. A Convenient Change of Measure

In discussing cases not covered by Theorem 2, it will be convenient toexpress certain backward SDEs in terms of a new probability, P� , definedas follows. Recalling that k is given by (11), we define the new probabilityP� through its density:

Et _dP�dP&#!t #exp \&

12 |

t

0(1&ks)

2 's } 's ds&|t

0(1&ks) '$s dBs+ . (12a)

(Since r and ' are assumed bounded, the right-hand side defines a mar-tingale and (12a) is therefore consistent.) The expectation operator withrespect to P� will be denoted E� , and the corresponding conditional expecta-tion given Ft will be denoted E� t . In particular, for any random variable Z,E� t[Z]=Et[!TZ]�!t . By Girsanov's theorem, the process B� , defined by

B� t=Bt+|t

0(1&ks) 's ds, (12b)

is n-dimensional standard Brownian motion under the measure P� .

5.2. The Case of Zero Gamma

Throughout this subsection, we assume that C and f are specified by (8)with #=0, and therefore kt=;[;&:(1&e&;(T&t))]&1.

In order to solve the first-order conditions, we introduce the auxiliaryprocesses (J, Z) # Dexp

0 _Dn (where Dexp0 is defined in Appendix A) as the

unique adapted solution to the backward SDE

dJt=&_(1&kt) \;&rt&kt

2't } 't++kt(:&;) Jt+

12

Zt } Zt& dt

+Zt dB� t , (13)

JT=0.


Existence and uniqueness of a solution are guaranteed by Theorem A1 ofAppendix A. Simple solutions are obtained if :=; or if :=0. Lemma A1implies that

exp(Jt)=E� t _exp \|T

t(1&ks) \;&rs&

ks

2's } 's ds++& , if :=;.

For :=0, the unique solution is the zero solution. The PDE characteriza-tion of (13) in a Markovian setting is discussed at the end of this section.

Lemma 3. Suppose that C and f are defined by (8) with #=0, and r and' are bounded. Then, given any *>0, the FBSDE system (4a), (4b) has aunique solution (X, V ) in D_V. The process X is given by

dXt=&_((;&:) kt&;) Xt+(:&;) Jt&;+rt+'t } 't

2 & dt&'$t dBt ,

with initial value X0=log(*), while V satisfies

1+:Vt=exp(Jt+(1&kt) Xt). (14)

In terms of the solution (X, V ) of Lemma 3, the optimal consumptionplan is given by

ct=I(t, Xt , Vt)=e&Xt(1+:Vt)=exp(Jt&ktXt). (15)

To complete the solution, we need to determine the value of *. Part (b) ofTheorem 2 implies that the optimal consumption-to-wealth ratio is

ct

Wt(c)=

;1&e&;(T&t) . (16)

In particular, c0=;w�(1&e&;T ), which together with (15) determines thevalue of * (and hence the initial value of X ). Applying Ito's lemma to (15)gives the dynamics of the optimal consumption plan (without having tosolve for * first):

dct

ct=+c

t dt+_ct dBt . (17)

Theorem 3. Suppose that C and f are defined by (8) with #=0, and rand ' are bounded. Then the dynamics of the (unique) optimal consumptionplan are given by (17), where

+ct =rt&;+_c

t } 't and _ct =kt '$t+Zt .


The optimal consumption rate as a fraction of wealth is given by (16), andthe corresponding optimal trading strategy is

�t=kt(_Rt _R$

t )&1 +Rt +(_R$

t )&1 Zt .

The first term of the optimal trading strategy is the instantaneouslymean-variance efficient strategy of Theorem 2(c). The second termrepresents the deviation from mean�variance efficiency due to the presenceof both a stochastic investment opportunity set, and time non-additivity.

5.3. The Case of Non-zero Gamma

Throughout this Section, we assume that C and f are given by (8) with#{0, and hence kt=k#(1&#(1+:))&1. The analysis is analogous to thezero gamma case, except that now the optimal consumption to wealth ratiois typically stochastic.

Applying Theorem A2 (of Appendix A), we uniquely define the(progressively measurable) processes (J, Z), where #J # D++

0 and �T0 Zt }

Zt dt<� a.s., as the solution to the backward SDE:

dJt=&_1#

(1+:)#�(1&#)+#

1&# \rt&;#

+k2

't } 't+ Jt+:k2

Zt } Zt

Jt & dt

+Zt dB� t , t<T, (18)

JT=JT&=0.

In the additive case, J is given by

Jt=1#

E� t _|T

texp \ #

1&# |s

t \ru&;#

+k2

'u } 'u+ du+ ds& ,

if :=0. (19)

The PDE characterization of (18) in a Markovian setting is discussed atthe end of this section.

Lemma 4. Suppose that C and f are defined by (8) with #{0, and r and' are bounded. Then, given any *>0, the FBSDE system (4a), (4b) has aunique solution (X, V ) in D_V. The process X is given by

dXt=&_:#

(1+:)#�(1&#) J &1t &(1+:) ;+rt+

't } 't

2 & dt&'$t dBt ,

with initial value X0=log(*), while V satisfies

Vt=(#�|#| ) |Jt|1+:k exp((1&k) Xt). (20)


In terms of (X, V ), the optimal consumption plan is

ct=I(t, Xt , Vt)=(1+:)1�(1&#) exp \&Xt

1&#+ |Vt|:k�1+:k

=(1+:)1�(1&#) exp(&kXt) |Jt|:k. (21)

Ito's lemma then delivers the optimal consumption dynamics.

Theorem 4. Suppose that C and f are defined by (8) with #{0, and rand ' are bounded. Then the dynamics of the optimal consumption plan aregiven by (17), where

+ct =

11&#

(rt&;)+_ct } 't+

k2 \

#1&#

't } 't&:

J 2t

Zt } Zt+ ,

_ct =k \'$t+

:Jt

Zt+ .

The optimal consumption-to-wealth ratio is given by

ct

Wt(c)=#&1(1+:)#�(1&#) J &1

t .

The optimal trading strategy is

�t=k(_Rt _R$

t )&1 +Rt +(1+:k)(_R$

t )&1 Zt

Jt.

As in the zero gamma case, the first term of the optimal portfolio expres-sion represents an instantaneously mean�variance efficient allocation, whilethe second term represents deviations from mean�variance efficiency due toa stochastic investment opportunity set. Formally setting :=&1 in theabove expression results in an instantaneously mean�variance efficientportfolio, for any price dynamics. This limiting case (not covered byTheorem 4) is revisited in Section 7.

5.4. PDE Characterization of (J, Z)

Given the above analysis, the agent's problem reduces to the computa-tionally well understood problem of solving the backward SDE that defines(J, Z). Below, we outline the PDE characterization of such a solution in aMarkovian setting.


In the remainder of this Section, we assume the Markovian structure ofcondition C5. We also recall the change of measure (12), and we use thedynamics of Y in the form dYt=+~ Y (t, Yt) dt+_Y (t, Yt) dB� t , where

+~ Y (t, y)=+Y (t, y)&(1&kt) _Y (t, y) '(t, y).

Letting Jt=h(t, Yt) for some function h, and using Ito's lemma, theBSDE characterizing (J, Z) leads naturally to a PDE, stated below forvarious parameter ranges. In all cases, the stated PDE can be viewed as asimplified version of (6) for the given parametric SDU class, obtained byletting

g(t, x, y)={:&1[exp((1&kt) x+h(t, y))&1],(#�|#| ) |h(t, y)| 1+:k exp((1&k) x),

if #=0;if #{0.

Given a solution, h, to the appropriate PDE, the BSDE solution is

Jt=h(t, Yt) and Zt=hy(t, Yt) _Y (t, Yt).

(In applications, one must of course confirm that (J, Z) are sufficientlyintegrable.)

We now state the PDE that has to be solved for all parameter ranges forwhich Theorem 2 does not provide a complete solution for general pricedynamics:9

Case 1 (#=0).

(;&:) kt h=(1&kt)(;&r& 12 kt'$')+ht+hy +~ Y

+ 12 tr[(hyy+h$y hy) _Y (_Y )$], h(T, } )=0. (22)

For :=#=0, this PDE is also valid, but its solution is zero, correspondingto the optimal portfolio allocation of Theorem 2(c). Another simplificationarises if :=;, as we discussed in Section 5.2.

Case 2 (#{0 and :{0).

&#

1&# \r&;#

+k2

' } '+ h=1#

(1+:)#�(1&#)+ht+hy+~ Y

+12

tr _\hyy+:kh

h$yhy+ _Y (_Y)$& , (23)

h(T, } )=0.


9 The notation tr(A), where A is any square matrix, stands for the trace of A, that is, thesum of its diagonal elements. For any matrices A, B such that the products AB and BA arewell defined, the identity tr(AB)=tr(BA) holds. Thus tr[h$yhy _Y (_Y)$]=hy_Y_Y$h$y andtr[hyy _Y (_Y)$]=tr[(_Y)$ hyy_Y].

Case 3 (#{0 and :=0).In this case, PDE (23) applies, but it can be further simplified. Let us

assume that the functions r(t, y) and '(t, y) are time independent, allowingus to simplify notation to rt=r(Yt) and 't='(Yt). (This is no less generalthan condition C5, since a time component can be added to Y.) UsingEq. (19), we have

h(t, Yt)=1# |

T

te p(s&t, Yt) ds

where

e p(s&t, Yt)=E� t exp _ #1&# |

s

t \ru&;#

+k2

'u } 'u+ du& .

The function p({, y) can be computed as the solution to the PDE

&#

1&# \r&;#

+k2

' } '+=&p{+ py+~ Y+

12

tr[( pyy+ p$y py) _Y (_Y)$], p(0, } )=0. (24)

6. EXAMPLES WITH `ÀFFINE'' DYNAMICS

For the homothetic SDU class of Section 4, we have seen in the lastsection that the agent's problem reduces to the solution of a single back-ward SDE, which in general can be solved numerically, for example, bynumerically solving an associated quasilinear PDE. The complexity of thelast step is a function of the price dynamics specification. For example, atrivial solution results if r and ' are deterministic. In this section we outlinea more general class of price dynamics, familiar from the term-structureliterature (see, for example, Duffie and Kan [15]), for which relativelysimple solutions to the first-order conditions result.

This section's models are not strictly special cases of Theorems 3 and 4,because r and ' are not bounded. For some parameter ranges, our exactproofs apply. For other parameter values, however, the utility process candiverge to infinity in finite time, and our verification arguments do notapply, because of integrability issues. Below, we provide some coarsesufficient conditions that keep utility finite, while in some additive casesa complete closed-form solution is possible (and optimality verificationis straightforward). A complete characterization, however, requires an


understanding of the behavior of the underlying affine dynamics that, toour knowledge, is not currently available in the literature and is beyond thescope of this paper.

Throughout the section, we assume the SDU specification of Section 4and that either #=0 or :=0. Under the price dynamics described below,we show that the relevant backward SDE reduces to a Riccati equationthat can either be solved in closed form or is straightforward to solvenumerically.

6.1. Price Dynamics

Given any v # Rn, diag(v) denotes the diagonal matrix with v as itsdiagonal, - v denotes the (column) vector (- v1 , ..., - vn )$, and v2 denotes(v2

1 , ..., v2n)$. Also, given any square matrix A, Aii denotes its i th diagonal

entry.We postulate a state process Y, valued in Rn, with dynamics

dYt=(%&}Yt) dt+7 diag(- &+`Yt ) dBt , (25)

where %, & # Rn and `, }, 7 # Rn_n. Y is assumed to be the unique processthat is valued in

Y=[ y # Rn : &i+ìy>0, for all i],

and satisfies (25). We refer to Duffie and Kan [15] and Dai et al. [9] forconditions on the parameters sufficient for this assumption to hold.

The price dynamics we consider are of the following two types:

Model A. rt=a+b } Yt and 't=diag(- &+`Yt ) 8, where a # R andb, 8 # Rn.

Model B. rt=ar+b$rYt+Y$tcr Yt and 't=a'+b'Yt , where ar # R,a' , br # Rn, and b' , cr # Rn_n, and cr is symmetric. Moreover, `=0 and&i=1 for all i.

Given the above assumption, we show below how to solve theappropriate PDE of Section 5.4. The proposed solution to the consumption-portfolio problem is then formally constructed as in Theorems 3 and 4.

6.2. The Case of Zero Gamma

In this Section, we assume that f is given by (8) with #=0.For Model A, we conjecture that PDE (22) has an affine solution,

h(t, y)=Gt+Ht } y,

where H and G are functions of time valued in Rn and R, respectively.


Direct computation shows that h solves (22) if G, H solve

H4 t=[(;&:) kt I+}$+(1&kt) `$ diag(8) 7$] Ht+(1&kt) _b+kt

2`$82&

&12

`$(7$Ht)2, HT=0,

G4 t=(;&:) ktGt&(1&kt) _;&a&kt

2&$82&

&H$t _%&(1&kt) 7 diag(&) 8+12

7 diag(&) 7$Ht& , GT=0.

The first equation is a Riccati equation in H, and although straightforwardto solve numerically, it can diverge to infinity in finite time. A sufficientcondition that precludes this is given by the following result:

Lemma 5. Suppose that b, `�0, and �i ìj>0 for all j. If :<0, then theRiccati equation in H has a finite solution for any finite T.

Given H, G is easily computed using the second equation. A closed formsolution can easily be obtained in the infinite horizon version of this model(T=�), if, for example, `, }, and 7 are diagonal.

For model B, direct computation shows that a solution to PDE (22) isgiven by

h(t, y)=Ct+D$t y+y$Ft y,

with F symmetric, and where C, D, and F are deterministic functions oftime, obtained by solving the following three equations, in order:10

F4 t=(1&kt) \cr+kt

2b$'b'++(;&:) kt Ft+[}+(1&kt) 7b']$ Ft

+Ft[}+(1&kt) 7b']&2Ft77$Ft , FT=0.

D4 t=[(;&:) ktI+}$&2F $t77$+(1&kt) b$'7$] Dt+(1&kt)[br+ktb$'a']

&2Ft[%&(1&kt) 7a'], DT=0,

C4 t=kt(;&:) Ct&(1&kt) _;&ar&kt

2a$' a'&

&D$t _%&(1&kt) 7a'+12

77$ Dt&&tr(Ft77$), CT=0.


10 We use the fact that y$Ay=0, \y, for some A # Rn_n if and only if A+A$=0.

Lemma 6. If cr is positive semidefinite and :<0, then the Riccatiequation in F has a finite solution for any finite T.

A closed form solution can easily be obtained in the infinite horizon ver-sion of this model (T=�), if, for example, cr , b' , 7, and } are diagonal.A simple solution (with finite T ) is also obtained if cr=b'=0, implyingthat r is an Ornstein�Uhlenbeck process and ' is constant. In this case Zand _R$

t � are deterministic, F=0, and

Dt=&\|T

te&}$(s&t)e&;(s&t)(k&1

s &1) ds+ ktbr .

6.3. The Additive Case

In this Section, we assume that f is given by (8) with :=0 and Case 3of Section 5.4 therefore applies. Since the solution method is analogous tothe #=0 case above, we briefly outline the approach, referring to Schroderand Skiadas [45] for further details.

For Model A, we conjecture a solution to (24) of the form p({, y)=G{+H{ } y, where H and G are functions of time valued in Rn and R,respectively. Direct computation shows that H and G satisfy an ODEsystem analogous to the corresponding system for #=0 above. Moreover,Lemma 5 (with analogous proof) remains valid if the assumptions #=0and :<0 are replaced with #<0 and :=0. A closed form solution (givenin Schroder and Skiadas [45]) is easily obtained if `, }, 7 are diagonal,bi , ìi , }ii�0, and ;>0.

For Model B, (24) is solved by a function of the form p({, y)=C{+D${ y+ y$F{y, with F symmetric, and where C, D, and F are deter-ministic functions of time satisfying an ODE system analogous to thecorresponding system for #=0 above. In many cases, the Riccati equationin F can be solved analytically using a procedure described by Gelb [27,Sect. 4.6]. Moreover, Lemma 6 (with analogous proof) remains validif the assumptions #=0 and :<0 are replaced with #<0 and :=0. Aparticularly simple solution is obtained if cr=b'=0, in which case F=0.A closed form solution (given in Schroder and Skiadas [45]) is also easilyobtained if cr , b' , 7, and } are diagonal, and }, cr�0.

7. INTRODUCING A BEQUEST FUNCTION

We conclude the main part of this paper with an outline of how to incor-porate a bequest function in our earlier analysis. We illustrate with aparametric example that has the interesting property that the optimal


portfolio is instantaneously mean�variance efficient under any pricedynamics, while the optimal consumption-to-wealth ratio is typicallystochastic. This type of solution has been discussed previously in a discrete-time setting by Giovannini and Weil [29]. We will keep our discussionshort by omitting existence and verification arguments, as well as mosttechnical details.

7.1. General Formulation

We modify our earlier setting, by endowing D with the new innerproduct

(x, y) =E _|T

0xt yt dt+xT yT& .

(In particular, x= y in D implies that xT= yT a.s.) Given consumptionplan c # C�D+, we interpret cT as a lump sum terminal consumption.

The utility process V(c) is defined as a unique solution to the backwardSDE

Vt(c)=Et _|T

tf (s, cs , Vs(c)) ds+v(cT)& , t # [0, T],

for some function v : R+ � R. We assume that the utility functionV0 : C � R is strictly increasing. The corresponding utility gradient can bewritten (under technical assumptions) as {V0(c; h)=(m(c), h) , where mt

is given by (3) for t<T, and

mT (c)=exp \|T

0fv(s, cs , Vs(c)) ds+ v* (cT),

where v* denotes the derivative of v.Taking as given an Arrow�Debreu state price density ? # D++, the

agent's problem is, as before, to maximize V0(c) subject to the budgetfeasibility constraint (?, c) �w. Assuming, for simplicity, that c # D++,the first-order condition for optimality of c is m(c)=*? for some *>0,together with the budget feasibility constraint. The solution method of thissystem is analogous to our earlier analysis, with the main new elementbeing the boundary condition corresponding to the fact that VT(c)=v(cT).We leave all details to the interested reader, and we proceed with anoutline of an interesting example.


7.2. A Homothetic Example

The parametric SDU form that we analyze in the remainder of thissection is

Vt=Et _|T

t \c#s

#exp(&#Vs)&

;#+ ds+log(cT)+

$# & , t # [0, T],

where ;�0, 0{#<1, and $ # R. The ordinally equivalent utility processV� t=#&1 exp(#Vt) satisfies the more suggestive recursion

V� t=Et _|T

te&;(s&t) \c#

s

#ds&

12

V� &1s d[V� ]s++e&;(T&t)+$ \c#

T

# +& ,

where [V] again represents the quadratic variation process of V. Apartfrom the bequest function (which can be made arbitrarily small by decreas-ing $), this corresponds to our earlier homothetic expression, (10), with#{0 and :=&1. (In Theorem 4 we assumed :>&1 in order to avoidhaving VT=&�, corresponding to $=&� in the above formulation.)

What is interesting about this case is that a consumption plan satisfyingthe first-order conditions of optimality can be financed by an instantaneouslymean-variance efficient portfolio (no matter what the state price dynamicsare). To see that, we note that, since fv=&cfc , the first-order conditionscan be rearranged to

&ddt

exp \|t

0fv(cs , Vs) ds+=*?tct and

exp \|T

0fv(cs , Vs) ds+=*?TcT .

It follows that the wealth process, W, corresponding to c satisfies

?t Wt=Et _|T

t?scs ds+?TcT&=

1*

exp \|t

0fv(cs , Vs) ds+ . (26)

Therefore, ?tWt is absolutely continuous, which is equivalent to thecondition _W='$. Arguing as in Section 2.3, it follows that the optimalportfolio allocation is the same as that of an investor with time-additivelogarithmic utility (given in Theorem 2(c) with kt=1).

The optimal consumption-to-wealth ratio is given by

ct

Wt=exp \&

#Jt

1&#+ , t # [0, T),


where the process J solves11

Jt=Et _|T

t

1&##

exp \&#Js

1&#+&;#

+rs+'s } 's

2ds+

$# & .

The PDE characterization of J in a Markovian setting, and examplesanalogous to those of Section 6 are left to the reader.

We conclude with an outline of a proof of the above claim, omittingseveral technical details. Let us first define the process X as the solution to(4a), with

Vt=Jt&Xt and I(t, x, v)={exp((x+#v)�(#&1)),exp(&x),

if t<T;if t=T.

With (X, V ) thus defined, we claim that the optimal consumption plan isgiven by

ct=I(t, Xt , Vt)={exp(&Xt&#Jt�(1&#)),exp(&XT),

if t # [0, T );if t=T,

(27)

where X0=log(*) has been chosen to make the budget constraint tight.(We will see later that *=&w.) To show that, suppose for the momentthat V is in fact the utility process, V(c), corresponding to c. Equation (27)is equivalent to fc(ct , Vt)=exp(Xt) for t<T, and v* (cT)=exp(XT). On theother hand, Eq. (4a) gives

exp(Xt)=*?t exp \&|t

0fv(cs , Vs) ds+ . (28)

This confirms the first-order condition m=*?, which (by an omittedverification argument) implies optimality. To confirm that V=V(c), wenotice that

& fv(ct , Vt)=#f (ct , Vt)+;=exp \&#Jt

1&#+ .

The drift term of J is therefore equal to &( fv+r+' } '�2)& f, whichtogether with (4a) implies that the drift term of dVt+ f dt vanishes. It


11 For the sake of notational simplicity, our choice of J here is not consistent with that inSection 5.3. It can easily be shown that J� t=exp(#Jt �(1&#)) is the first component of thesolution to a backward SDE analogous to (18).

follows that dVt=&f (ct , Vt) dt+dMt , for some martingale M, and sinceVT=($�#)&XT=VT (c), V=V(c).

Finally, Eqs. (26) and (28) give Wt=exp(&Xt) (and therefore *=&w).Combining this with (27) results in the claimed consumption-to-wealthratio formula. The optimal consumption dynamics can also be easilyobtained by applying Ito's lemma to (27).

APPENDIX A

A Class of Backward SDE

In this appendix we prove existence, uniqueness, monotonicity, andconvexity properties for a class of backward SDE that arise in connectionwith the homothetic specification of SDU discussed in Section 3.

The backward SDEs we will consider are of the form

dVt=&(Ut&;tVt+12A(Vt) &Zt&2) dt+Zt dBt , VT=0, (A1)

to be solved for an adapted pair (V, Z). Theorem 1 and the analysis ofSection 5 require the solution of the above backward SDE with either A(Vt)or A(Vt)Vt being a deterministic constant (and proper restrictions on U and;). We now consider these two cases in turn. The terminal value VT has beenset to zero for brevity of exposition and since this is the case we focus on inthe main text. The mathematical arguments, however, extend readily toincorporate a more general terminal value. In fact, in some cases it willbecome apparent that the zero-terminal-value case is mathematically themost difficult one, due to integrability concerns as time approaches T.

Throughout this appendix, the underlying probability space, then-dimensional Brownian motion, B, and associated filtration, and the spaceD (with its positive cone D+ and strictly positive cone D++) are as inSection 2. Dn denotes the Cartesian product of n copies of D. For anyZ # Dn, each value, Z(|, t), should be thought of as a n-dimensional rowvector with Euclidean norm &Z(|, t)&. (In particular, & }& does not denotethe norm induced by the inner product of D.) In addition to the set D0 ofSection 4.2, the following subsets of D will be used:

D1={X # D : E _|T

0|Xt|

l dt&<�, for every l # (0, �)= ,

Dexp0 =[X # D : E[exp(ess sup

tl |Xt| )]<�, for every l # (&�, �)].

Dexp1 ={X # D : E _exp \l |

T

0|Xt| dt+&<�, for every l # (&�, �)= .


For any S/D, we define S+=S & D+ and S++=S & D++. As always,we identify processes that are modifications of each other.

The Case of Constant A(V )

This section is on the backward SDE (A1) under the assumption that Ais identically equal to some constant :, U # Dexp

1 , and ; # D is non-negativeand bounded. For :=0, one obtains easily the solution: Vt=Et[�T

t Us exp(&�s

t ;u du) ds]. Assuming :{0, a simple rescaling shows that we canassume without loss of generality that :=1. Our objective in this sectionis therefore to analyze the backward SDE

dVt=&(Ut&;tVt+12 &Zt&2) dt+Zt dBt , VT=0. (A2)

The following is the central result of this section.

Theorem A1. Suppose that U # Dexp1 , and ; # D+ is non-negative and

bounded. Then there exists a unique pair (V, Z) # Dexp0 _Dn satisfying (A2).

Moreover, the solution V as a function of the parameter U is monotonicallyincreasing and convex.

The proof of Theorem A1 will proceed in a sequence of lemmas. Weassume throughout that U # Dexp

1 and that, for some ;� # R, 0�;t�;� for all t.We begin with a convenient reformulation of the problem. For any

(V, Z) # Dexp0 _Dn and stopping time {, we consider the recursion

Vt=log \Et _exp \|{

tUs&;s Vs ds+V{+&+ on [{�t]. (A3)

Lemma A1. The following statements are equivalent, for any V # Dexp0 :

(a) There exists a unique Z # Dn such that (V, Z) satisfies (A2).

(b) There exists some Z # Dn such that (V, Z) satisfies (A2).

(c) V satisfies (A3) with {=T, and VT=0.

(d) V satisfies (A3) for any stopping time {, and VT=0.

Proof. (d) � (c). Clearly, (c) is implied by (d). Conversely, supposethat (c) holds. We then have

exp(Vt)=exp \&|t

0Us&;sVs ds+ Mt , (A4)

where M is the martingale defined by

Mt=Et _exp \|T

0Us&;sVs ds+& . (A5)


Suppose now that { is any stopping time. By the optional samplingtheorem, Mt=Et[M{] on [{�t]. From (A4), we have

exp \|{

tUs&;sVs ds+V{+=exp \&|

t

0Us&;s Vs ds+ M{ on [{�t].

Applying the operator Et on both sides, simplifying the right-hand side,and using (A4) again, (d) follows.

(c) O (b). Suppose that (c) holds, and let M be the martingaledefined by (A5), so that (A4) is also valid. By the martingale representationtheorem, there exists a unique predictable Z� # Dn such that dMt=Z� t dBt .Let the process Z be defined by Zt=Z� tM &1

t . Suppose, for now, thatZ # Dn. Letting Wt=exp(Vt), and applying integration by parts to (A4),we obtain

dWt

Wt=&(Ut&;t Vt) dt+Zt dBt . (A6)

Finally, (A2) follows from an application of Ito's lemma to Vt=log(Wt).To conclude that (b) holds, however, we still need to show that Z # Dn,

which we do next. Let Nt=M &1t and N*=maxt Nt . From the definition

of Z and the Cauchy�Schwarz inequality, we have

\E _|T

0&Zt&2 dt&+

2

�\E _(N*)2 |T

0&Z� t&2 dt&+

2

�E[(N*)4] E _\|T

0&Z� t&2 dt+

2

& .

To show that Z # Dn, it suffices to show that the above expression is finite.By the Burkholder�Davis�Gundy inequalities, �T

0 &Z� t&2 dt has a finitesecond moment if MT has a finite fourth moment, which it does sinceV # Dexp

0 and U # Dexp1 . To show that N* has finite fourth moment, we first

notice that, by Jensen's inequality, N is a submartingale. By Doob'smaximal inequality and (A5), there exists some constant C such that

E[(N*)4]�CE[N 4T]=CE _exp \&4 |

T

0Us&;sVs ds+& .

The latter is finite since V # Dexp0 and U # Dexp

1 , and the proof that Z # Dn

is complete.

(b) O (c). Suppose that (V, Z) # Dexp0 _Dn satisfies (A2), and define

Wt=exp(Vt). By Ito's lemma, W satisfies (A6). Let M be the stochastic


exponential of Z, that is, the unique local martingale that satisfiesdMt=Mt Zt dBt , M0=1. (As is well known, stronger restrictions on Zthan mere membership to Dn are required to make M a martingale.)Integration by parts then gives (A4). Given any stopping time {, we letM {

t =Mt1[t�{]+M{1[t>{] . Let [{(n)] be an increasing sequence of stop-ping times that converges to T, such that the stopped process M {(n) is amartingale, for every n. Arguing as in the first part of the proof (where (A3)was derived from (A4) and the martingale property of M), we have

exp(Vt)=Et _exp \|{(n)

tUs&;sVs ds+V{(n)+& on [{(n)�t].

Letting n � � and using the dominated convergence theorem (madepossible by the assumption V # Dexp

0 and U # Dexp1 ), (c) follows. Having

shown (c), it follows that M is given by (A5), and it is after all a truemartingale.

(b) � (a). Clearly, (a) implies (b). Suppose that (b) holds, and letWt=exp(Vt). Then Z represents the diffusion term of the Ito decomposi-tion of dWt�Wt and is therefore uniquely determined in D. K

An interesting corollary of Lemma A1 is that for ;=0, (A3) (with {=T )gives a closed-form expression for the V # Dexp

0 that is part of the uniquesolution to (A2). For the case of nonzero ;, however, a closed-form expres-sion is not apparent to us, and we will resort to a fixed-point argument.For this purpose, we define a function FU : Dexp

0 � Dexp0 , corresponding to

the parameter process U, as follows:

FU (V )t=log \Et _exp \|T

tUs&;sVs ds+&+ , t # [0, T].

(That FU is indeed valued in Dexp0 can easily be confirmed using Doob's

maximal inequality.) By Lemma A1, we are interested in finding a uniquefixed point of the function FU and in showing the monotonicity of the fixedpoint as a function of the parameter U. Assuming existence for now,uniqueness and monotonicity will be consequences of the following result:

Lemma A2. Suppose that, for some U # Dexp1 , the following conditions

hold:

(a) V # Dexp0 satisfies V=FU (V ).

(b) V� # Dexp0 is continuous, V� T�0, and, for any stopping time {,

V� t�log \Et _exp \|{

tUs&;s V� s ds+V� {+&+ on [{>t], t # [0, T].


Then V� t�Vt , for all t # [0, T]. The result also holds with all inequalitiesreversed.

Proof. Suppose that, for some t, the event A=[V� t<Vt] is non-null.Consider the stopping time

{=inf[s�t : V� s�Vs].

Since almost all paths of V� and V are continuous and V� T�0=VT , wehave V� {=V{ on A, while V� s<Vs on A & [t�s<{]. By our hypothesesand Lemma A1, we have, on event A,

0>exp(V� t)&exp(Vt)

�Et _exp \|{

tUs&;sV� s ds+V� {+

&exp \|{

tUs&;sVs ds+V{+&�0,

a contradiction. The same argument applies with all inequalitiesreversed. K

Suppose now that, for each i # [1, 2], U i # Dexp1 , and V i # Dexp

0 satisfiesFUi (V i)=V i. An immediate consequence of the last two lemmas is that ifU1�U2, then V1�V 2, proving the monotonicity claim of Theorem A1.Moreover, taking U1=U2=U, it follows that any two fixed points, V1 andV2, of FU must satisfy both V1�V2 (since U1�U2) and V2�V 1 (sinceU2�U1). Therefore V1=V2, proving that FU has at most one fixed point.

Next, we turn to the question of existence of a fixed point of FU . We beginwith the special case of a bounded U. Let B be the space of progressivelymeasurable bounded processes, metrized by the (pseudo)metric

d(x, y)=ess sup(|, t)

|x(|, t)& y(|, t)|, x, y # B.

As usual, we identify any two processes x, y # B such that d(x, y)=0,which makes (B, d ) a complete metric space. B is ordered in the usualsense: x� y means P[xt� yt]=1 for all t.

Lemma A3. If U # B, then FU (V )=V for some V # B.

Proof. Since U is fixed throughout, we simplify notation by lettingF=FU . We use Blackwell's version of the contraction fixed-point theorem(see, for example, Theorem 3.3 of Stokey and Lucas [48]) to show that Fis a contraction if ;� T<1 (recall that ;� is an upper bound of ;). This partial


result will then be generalized by partitioning the time horizon and by piec-ing together a solution backward in time. Alternatively, for any T, we canshow that F composed with itself k times is a contraction for sufficientlylarge k.

Given any real number x, we denote also by x the function in B identi-cally equal to x. From the functional form of F it follows easily that, forevery V # B and positive real x, F(V+x)�F(V )&;� Tx. Consider now anyV, W # B. Since V�W+d(V, W ) and F is decreasing, we have F(V )�F(W+d(V, W ))�F(W )&;� Td(V, W ). Interchanging the roles of V andW, we also have F(W )�F(V )&;� Td(W, V ). Therefore, d(F(V ), F(W ))�;� Td(V, W ), proving that F is a contraction if ;� T<1.

If ;� T�1, choose any integer N>;� T. Initially, set k=N&1. The aboveargument shows that there exists a V # B satisfying (A2) for all t�(k�N) T.We proceed inductively. Suppose that the last statement is true forarbitrary k # [1, ..., N&1]. Let V # B be a solution of (A2) over the timehorizon [Tk�N, T]. Applying the same contraction argument over the timeinterval [T(k&1)�N, Tk�N] with terminal value VkT�N , the proof of thelemma is easily completed.

Alternatively, we can define dt(x, y)=ess sup(|, s�t) |x(|, s)& y(|, s)|.Then, by the argument used above, dt(F(V ), F(W ))�;� (T&t) d0(V, W ).A similar argument repeated k times gives

dt(F (k)(V ), F (k)(W ))�(;� (T&t))k

k !d0(V, W ).

For large enough k, F (k) is therefore a contraction and, for a unique V # B,F (k)(V )=V. Applying F on both sides of this equation, it follows thatF(V ) is also a fixed point of F (k) and is therefore equal to V. This showsthat V is a fixed point of F. K

Using the last two lemmas, we now show that, for any U # Dexp1 , FU has

a fixed point in Dexp0 . Suppose first that U # Dexp

1 is bounded below. Forevery integer n, let U n

t =min[Ut , n], and let Vn # B solve FUn(Vn)=Vn. Byour earlier results, Vn exists and is monotonically increasing in n. Usingthese facts, we have

V nt �log Et _exp \|

T

tUs&;s V 1

s ds+& .

Since V1 # Dexp0 , it follows that the sequence [Vn] is bounded above almost

surely, and therefore there exists some V such that Vn A V a.s. as n � �.Using Doob's maximal inequality, it is also easy to conclude from the


above bound on Vn that V # Dexp0 . Letting n go to infinity in FUn(Vn)=V n

and using the dominated convergence theorem, we have FU (V )=V. Thisproves the lemma assuming that U is bounded below. To extend the exist-ence proof for U unbounded below, we use an analogous argument withUn=max[U, &n], showing that the corresponding solutions Vn convergemonotonically from above to a fixed point of FU . This completes the proofof existence and monotonicity.

Finally, we prove the convexity of the solution V as a function of theparameter U. For i # [a, b], let U i # Dexp

1 , and let (V i, Z i) # Dexp0 _Dn

satisfy

dV it=&(U i

t&;tV it+

12 &Z i

t&2) dt+Z i

t dBt , V iT=0.

Fixing an arbitrary & # (0, 1), we define (V &, Z&)=&(Va, Za)+(1&&)(Vb, Zb), and we assume that (V, Z) # Dexp

0 _Dn is the solution to (A2)corresponding to U#&Ua+(1&&) Ub. Convexity follows, if we can provethat V &

0�V0 . To this end, we define 22t=(& &Zat &

2+(1&&) &Zbt &

2)&&Z&

t&2>0 (positivity follows from Jensen's inequality). It follows that

dV &t =&(Ut+2t&;tV &

t + 12 &Z&

t&2) dt+Z&t dBt , V &

T=0.

Ignoring for now the need for an integrability restriction on 2, Lemma A1implies that, for any stopping time {,

exp(V &t )=Et _exp \|

{

tUs+2s&;sV &

s ds+V &{+&

>Et _exp \|{

tUs&;s V &

s ds+V &{+& .

By Lemma A2, we obtain V &0�V0 (and in fact, it is not hard to show that

the inequality is strict).As noted above, the application of Lemma A1 is not justified without

appropriate integrability restrictions on 2, a difficulty that is easily over-come by slightly modifying Lemma A1. Returning to the last part of theproof of Lemma A1 ((b) O (c)), we make the following two changes: (i)We assume the localizing sequence [{(n)] converges to a stopping time {,instead of T (by just taking the minimum of the original sequence elementsand {). (ii) By replacing 2 with min[2, 1] before taking expectations, weconclude that

exp(V &t )�Et _exp \|

{(n)

tUs+min[2s , 1]&;sV &

s ds+V &{(n)+& ,

n=1, 2, ...


Taking the limit as n � �, we can now complete the proof as before withoutthe integrability concern. This also completes the proof of Theorem A1.

The Case of Constant A(V ) V

In this section we analyze the backward SDE (A1) with the specificationA(v) v equal to some constant greater than or equal to minus one,U # D++

1 , and ; # Dexp1 . Making the change of variable V� t=Vt exp

(&�t0 ;s ds), it becomes clear that we can assume without loss in generality

that ;=0, which we do. More precisely, we consider the backward SDE

dVt=&\Ut+p&1

2&Zt&2

Vt + dt+Zt dBt , t<T, VT=VT&==, (A7)

where =�0, p>0, and U # D++1 . Although for the purposes of the main

text we are only interested in the case ==0, we will attack this case by firstsolving with a positive =, and then letting = approach zero.

Let L2 be the space of all progressively measurable processes, Z, valuedin Rn, satisfying �T

0 &Zt&2 dt<� a.s. As usual, we identify any two elementsZ, Z� # L2 such that �T

0 &Zt&Z� t&2 dt=0 a.s. The following is the section'smain conclusion

Theorem A2. Suppose that = # [0, �), p # (0, �), and U # D++1 . Then

there exists a unique pair (V, Z) # D++0 _L2 satisfying (A7). Moreover, the

solution V as a function of the parameter U is monotonically increasing, andit is convex if p # [1, �) and concave if p # (0, 1].

In the remainder of this section we prove Theorem A2. The structure ofthe proof parallels that of Theorem A1, but the details are considerablymore delicate. We assume throughout that =�0 and p>0.

We begin with the closely related recursion

V pt =Et _|

{

tpUsV p&1

s ds+V p{& on [{>t], (A8)

where { is any stopping time.

Lemma A4. Suppose that U # D++1 and V # D++

0 . Then the followingconditions are equivalent:

(a) There exists a unique Z # L2 such that (V, Z) satisfies (A7).

(b) There exists some Z # L2 such that (V, Z) satisfies (A7).

(c) V satisfies (A8) with {=T and VT==.

(d) V satisfies (A8) for any stopping time {, and VT==.


Proof. The key to the proof lies in the fact that (A7) together with Ito'slemma implies

dV pt =&V p&1

t pUt dt+ pV p&1t Zt dBt .

The details are analogous to Lemma A1. The weaker integrability restric-tion on Z simplifies part of the proof. On the other hand, in deriving theintegral representation from the differential representation, it is necessaryto use a localizing stopping time sequence and take a limit (usingmonotone convergence for the first term, and dominated convergence forthe second). The details are left to the reader. K

For every =�0 and U # D++1 , we define the operator F =

U : D++0 � D++

0 by

F =U (V )t=\Et _|

T

tpUsV p&1

s ds+= p&+1�p

.

That F =U is indeed valued in D++

0 is a consequence of Doob's maximalinequality. Because of Lemma A4, we are interested in proving that F =

U hasa unique fixed point. We will prove this, as well as the claimedmonotonicity and convexity properties of the fixed point, for =>0 first, andwe will then let = approach zero.

We start with two lemmas that are used below, as well as in the proofof Theorem 4.

Lemma A5. Suppose that V=F =U (V ) for some = # [0, 1] and U # D++

1 .Then, there exists some positive constant K, such that

Et _|T

tUs ds&�Vt�K \Et _|

T

tU p

s ds+1&+1�p

if p # (1, �),

and

K \Et _|T

tU p

s ds&+1�p

�Vt�Et _|T

tUs ds+1& if p # (0, 1).

For p=1, we have Vt=Et[�Tt Us ds].

Proof. The case p=1 is immediate from Lemma A4. Suppose now thatp>1. By Lemma A4,

Vt=Et _|T

t \Us ds+p&1

2(dVt)

2

Vt ++=&�Et _|T

tUs ds+=& .


To derive the right-hand inequality, we let \=( p&1)�p # (0, 1) and use thegradient inequality x\�1+\(x&1), x # R. In particular, lettingx=(Vt �Ut)

p and rearranging, we obtain pUt V p&1t �U p

t +( p&1) V pt .

Therefore,

V pt =Et _|

T

tpUsV p&1

s ds+= p&�Et _|T

tU p

s +( p&1) V ps ds+= p& .

Lemma C1 implies

V pt �Et _|

T

te( p&1)(s&t)U p

s ds+= pe( p&1)(T&t)& .

Finally, for p<1, all the above inequalities are reversed. K

Lemma A6. Suppose that U # D++1 , V, V� # D++

0 , V� is continuous,=, =~ # R, and if p # (1, �), then =~ >0. If V=F =

U (V ), V� T==~ �=�0, and, forany stopping time {,

V� t�\Et _|{

tpUsV� p&1

s ds+V� p{&+

1�p

on [{>t], t # [0, T],

then V� t�Vt for all t # [0, T].

Proof. Suppose first that p # (0, 1). Utilizing the gradient inequalityapplied to the convex function x [ x( p&1)�p, we obtain, for any stoppingtime {,

V� pt &V p

t �Et _|{

tpUs(V� p&1

s &V p&1s ) ds+V� p

{ &V p{ &

�Et _|{

t( p&1)

Us

Vs(V� p

s &V ps ) ds+V� p

{ &V p{ & .

The result now follows from Lemma C2. The proof is immediate for p=1.Finally, we consider the case of p>1. Again using the gradient inequalityapplied to the now concave function x [ x( p&1)�p, we obtain, for anystopping time {,

V pt &V� p

t �Et _|{

t( p&1)

Us

V� s

(V ps &V� p

s ) ds+V p{ &V� p

{& ,

and the result follows from Lemma C3. The required integrability restrictionin applying Lemma C3 is satisfied, because U # D1 and V� �=~ >0. K


The above lemma immediately implies that if U1, U2 # D++1 , F =i

Ui (V i)=V i for i # [1, 2], and (U1, =1)�(U2, =2)>(0, 0), then V1�V2. It alsoimplies that F =

U has at most one fixed point in D++0 if U # D++

1 and =>0.Next we argue that to prove Theorem A2 it suffices to show the case of

positive terminal value. To see that, suppose that V[=] # D++0 is the

unique fixed point of F =U for =>0, and the dependence of V[=] on U is

monotone and convex (monotonicity and uniqueness were proved in thelast paragraph). Recalling that V[=] is decreasing in =, we define Vt=lim= a 0 V[=]t . By Lemma A5 and dominated convergence, V # D++

0 is afixed point of F 0

U . Moreover, V inherits the monotone and convexdependence on U. It remains to show that V is the unique fixed point ofF 0

U . For p # (0, 1] this is immediate from Lemma A6, but the argumentdoes not apply with p>1 (because the integrability restrictions required byLemma C3 are violated). Instead, the following indirect argument applies.Suppose that V� is another fixed point of F 0

U . For any =>0, Lemma A6shows that V[=]�V� . Letting = approach zero proves that V�V� . To showthe reverse inequality, we confirm that V�V� += for any =>0. Fixing =>0,let V� =V� +=. Applying Ito's lemma to V� p (as in Lemma A4), it follows thatthe assumption of Lemma A6 is satisfied and V� �V. This completes theproof of uniqueness.

Given the above arguments, we assume that =>0 throughout theremainder of this proof. We first show that F =

U has a fixed point. Becauseof homogeneity, it is sufficient to prove the F 1

U has a fixed point. Weconsider the bounded case first

Lemma A7. Suppose that there exist constants k, K such that0<k<Ut<K, t # [0, T]. Then F 1

U has a fixed point in D++0 , which is also

bounded from above and away from zero.

Proof. We let the metric space of progressively measurable boundedprocesses (B, d ) be defined as in the last section, and we introduce thefollowing change of variables:

Xt=log( pUt), Yt= p log(Vt), \=p&1

p<1.

The lemma's hypothesis implies that X # B. The fixed-point conditionV=F 1

U (V ) can be written as

Yt=log \Et _|T

texp(Xs+\Ys) ds+1&+ , t # [0, T]. (A9)

The lemma will be proved if we can construct a Y # B satisfying (A9). For\ # (&1, 1), Blackwell's theorem (see, for example, Theorem 3.3 of Stokey


and Lucas [48]) applies to the operator of Eq. (A9) directly, proving thelemma for this case.

Next we prove by induction the following statement, for any integer k:If 1>\> &k, then there is a unique solution Y # B to (A9), for any X # B.For k=1, we have already proved the result. Suppose now we have shownthe result for some k, and we wish to prove it under the assumption&k�\>&k&1. We break the fixed-point problem into two parts

Yt=Zt=log \Et _|T

texp(Xs&(1&=) Ys+(\+1&=) Zs) ds&+1+ ,

where = # (0, 1) is chosen small enough so that \+1&=>&k. By theinduction hypothesis, for any choice of Y # B, there exists a correspondingZ # B that solves the second of the two equations. We let Z(Y ) denote theresulting functional relationship, which is monotonically decreasing.(Monotonicity can be shown by the same argument, following Lemma A5,that we used to prove the monotone dependence of the fixed point ofF =

U on U.) We complete the proof by showing that &Z( } ) satisfies theassumptions of Blackwell's theorem and therefore has a fixed point in B.

Let $=(1&=)�(\&=) # (&1, 0). Since Z( } ) is decreasing, it suffices toshow that, for any given constant process x�0, Z(Y+x)�Z(Y )+$x, or,equivalently, 2�$, where 2=(Z(Y+x)&Z(Y ))�x�0. Let

At=Et _|T

texp(Xs&(1&=) Ys+(\+1&=) Zs(Y )) ds& .

From the definitions of Z, 2, and $, we have

(At+1) exp(2tx)=At exp((\&=)(2t&$) x+2tx)+1

�At exp((\&=)(2t&$) x+2tx)+exp(2t x).

Canceling out the term exp(2t x) and then the factor At , it follows that(\&=)(2t&$)�0, and therefore 2t�$, completing the lemma's proof. K

To complete the existence proof, we need to show that F 1U has a fixed

point, without assuming that U is bounded from below and above.Suppose first that U is only bounded above, and consider the sequenceUn=max[1�n, U ], n # [1, 2, ...]. Let Vn be the fixed point of F 1

U n , shownto exist in Lemma A7. We have argued (with Lemma A6) that Vn+1�V n

for all n, and therefore Vt=limn � � V nt is well defined for all t # [0, T].

Moreover, the bounds of Lemma A5 imply that Vt>0 for t<T.Dominated convergence shows U is a fixed point of F 1

U . For general U, weconsider the sequence Un=min[n, U], and we apply a similar argument toconstruct a fixed point of F 1

U . This completes the proof of existence.


The argument that proves the convexity claims is completely analogousto the case of A(V )=0. Where we used the convexity of the functionz [ &z&2, one should now use the convexity of the function (z, v) [ &z&2�v,for v>0, and where we used Lemma A2, one should now use Lemma A6.This completes the proof of Theorem A2.

APPENDIX B

Proofs Omitted from Main Text

Proof of Lemma 1

Let 2t (&)=Vt (c+&h)&Vt (c), &�0, and

Lt (c)=Et _|T

texp \|

s

tfv(w) dw+ fc(s) hs ds& .

For any stopping time {, it follows that

Lt (c)=Et _|{

tfc(cs , Vs(c)) hs+ fv(cs , Vs(c)) Ls(c) ds+L{(c)& ,

while recursion (2) and the gradient inequality imply that

2t (&)&

�Et _|{

tfc(cs , Vs(c)) hs+ fv(cs , Vs(c))

2s(&)&

ds+2{(&)

& & .

The proof is completed by applying Lemma C3, with x=2(&)�&&L(c).

Proof of Theorem 1

The following proof relies heavily on the mathematical results ofAppendix A.

Case 1 (#=0). Given any c # C, we define the process Ut=: log(ct).Using the inequality exp( |x| )�exp(x)+exp(&x) and Jensen's inequality,it follows that U # Dexp

1 , and by Theorem A1, there exists a unique pair(V� , Z) # Dexp

0 _Dn such that

dV� t=&(Ut&;V� t+12 &Zt&2) dt+Zt dBt , V� T=0.


Defining the new process V to satisfy 1+:Vt=exp(V� t) and applying Ito'slemma to the above equation, we obtain

dVt=&(1+:Vt) \log(ct)&;:

log(1+:Vt)+ dt

+exp(V� t) Zt dBt , VT=0.

The second term of the above expression is a martingale, because

E _\|T

0exp(2V� t) &Zt&2 dt+

1�2

&<�,

a condition that can be easily confirmed using the Cauchy�Schwarzinequality and the fact that (V� , Z) # Dexp

0 _Dn. Integrating the expressionfor dVt from t to T and taking the conditional expectation Et , it followsthat V solves (2) when f is defined by (8) and #=0. Reversing the aboveargument also shows that V is the unique solution to (2) within the spaceV=[V : 1+:V # D++

0 ]. Monotonicity of V0 follows from the monotonedependence of V� on U, part of the conclusion of Theorem A1.

To show concavity, suppose first that 0�:�;. Then f is concave on thedomain [(c, v) : c>0, 1+:v>0]. (This is just a matter of confirming thatfcc<0 and fcc fvv� f 2

cv .) We can then apply Lemma 1 to show a gradientinequality that implies concavity. If :<0, then, by Theorem A1, theprocess V� in the above construction is convex and increasing in U, whichis in turn convex in c. This shows that 1+:V=exp(V� ) is convex in c, andtherefore V is concave in c.

Finally, homotheticity is easily discernible in the ordinally equivalentutilities defined in (9), since, for any *>0, we have

V� (*c)=V� (c)+log(*)1&exp(&;(T&t))

;.

This can easily be derived from (10) in differential form, which in turnfollows from the original recursion (2) and Ito's lemma.

Case 2 (#{0). Given any c # C, we define the process Ut=(c#t �|#| )

exp(&;t). By Theorem A2 and Lemma A4 (with p=1+:), there exists aunique V� # D++

0 such that

V� 1+:t =Et _|

T

t(1+:) UsV� :

s ds& , t # [0, T].


Suppose now that V is related to V� by Vt=(|#|�#) V� 1+:t exp((1+:) ;t).

Then the above recursion is equivalent to

Vt e&;(1+:) t=Et _|T

t(1+:)

c#s

#e&(1+:) ;s |Vs|

:�(1+:) ds& , t # [0, T].

Writing the above as

d(Vte&;(1+:) t)=&(1+:)c#

t

#e&(1+:) ;t |Vt |

:1+: dt+Zt dBt ,

for some Z # Dn, and using integration by parts, it follows that V solves (2)when f is defined by (8) and #{0. Letting V=[V # D0 : #V>0], the aboveargument shows that V is the unique solution in V. Monotonicity of V0 isalso a consequence of the above argument and Theorem A2.

To prove concavity of V0 , we distinguish cases. If :#>0, and, as usual,#<min[1, (1+:)&1], then f is concave on the domain [(c, v) : c>0,#v>0]. (This is just a matter of confirming that fcc<0 and fcc fvv� f 2

cv .)Lemma 1 can then be used to show a gradient inequality, which in turnimplies concavity. If :>0 and #<0, then by Theorem A2, V� in the aboveconstruction is convex and increasing in U, which is in turn convex in c.It follows that #V is convex in c, and therefore V is concave in c. If :<0and #>0, then V� , and hence V, is concave and increasing in U, which isconcave in c. This completes the proof of concavity.

Finally, homotheticity is easily discernible in the ordinally equivalentutility defined in (9), since, for any *>0, we have V� t (*c)=*#V� t (c).

Proof of Lemma 2

When #=0 and :>0, or :, #<0, the result follows from Lemma 1.The remaining cases follow (in some cases, using weaker integrabilityrestrictions than that assumed by Lemma 2).

Case 1 (#=0 and :<0). Let Yt=1+:Vt (c), Y� t=1+:Vt (c), 2c=c&c, 2Y=Y&Y� , f (t)= f (ct , Vt (c)), and analogously for the partials of f.By Lemma A1, we have

Y� t=Et _exp \|T

t: log(cs)&; log(Y� s) ds+& , (B1)

and similarly for Y. These conditions, together with the gradientinequalities

: log(c)�: log(c)+: 2c

c, and &; log(Y )�&; log(Y� )&;

2Y

Y�,


imply

2Yt�Et _exp \|T

t: log(cs)&; log(Y� s) ds+

_\exp \|T

t

:2cs

cs

&;2Ys

Y� s

ds+&1+& .

Letting Rt=exp(� t0 fv(s)+; ds) and using the fact that : log(cs)&

; log(Y� s)= fv(s)+; and the inequality ex&1�x, we obtain

2Yt�Et _RT

Rt\|

T

t

: 2cs

cs

&;2Ys

Y� s

ds+& .

Next we define the new probability, Q, by letting dQ�dP=RT �E[RT].Using the change of measure formula for conditional expectations and(B1), we have Et[dQ�dP]=Rt Y� t�E[RT], and hence

E Qt _|

T

t

: 2cs

cs

&;2Ys

Y� s

ds&=1

RtY� t

Et _RT \|T

t

:2cs

cs

&;2Ys

Y� s

ds+&�

2Yt

Y� t

.

By Lemma C1, it follows that

2Yt

Y� t

�:E Qt _|

T

te&;(s&t) 2cs

cs

ds&=:

RtY� t

Et _RT |T

te&;(s&t) 2cs

cs

ds& .

Simplifying, we obtain

Vt (c)&Vt (c)�1

RtEt _RT |

T

te&;(s&t) 2cs

csds&

=1

RtEt _|

T

tEs[RT] e&;(s&t) 2cs

csds&

=Et _|T

te&;(s&t) Rs

Rt

Y� s

cs2cs ds&

=Et _|T

texp \|

s

tfv(u) du+ fc(s) 2cs ds& .


Case 2 (#>0 and :>0). In this case, we show C2$ under the weakercondition E[�T&=

0 f 2v(cs , Vs(c)) ds]<� for every = # (0, T).

We fix some h # F(c) and define h=t=ht1[t # [0, T&=]] . Concavity of f, the

gradient inequality, the above restriction on fv , and Lemma C3 imply

V0(c+h=)�V0(c)+(m(c), h=) , = # (0, T). (B2)

The next step is to take the limit in (B2) as = a 0. The first-order conditions(1) and square integrability of h and ? imply lim= a 0 (m(c), h=) =(m(c), h) . The proof is completed by showing that lim= a 0 V0(c+h=)=V0(c+h). The dominated convergence theorem (justified below), con-tinuity of f in its arguments, and lim= a 0 h==h, imply

lim= a 0

Vt (c+h=)=Et _|T

tf (cs+hs , lim

= a 0Vs(c+h=)) ds& , t # [0, T].

If follows from Theorem A2 that lim= a 0 Vt (c+h=)=Vt (c+h). The last stepis to justify the interchange of limit and expectation. With the uniform (in=) bounds

ct �2� ct+h=t�2ct+ht

(because c+h>0, we can always rescale h to ensure that the lower boundholds),

0� f (ct+h =t , V(c+h=))� f (2ct+ht , V(2c+h)).

It is easy to show that c�2, 2c+h # C, which establishes the integrableuniform bounds.

Case 3 (#>0 and :<0). We prove C2$ under the restrictionE[exp(&3 �T&=

0 fv(cs , Vs(c))) ds]<� for every = # (0, T ).

We fix some feasible direction h # F(c) and define h= as in Case 2. Forany &�0, we define 2t (&)=Vt (c+&h=)&Vt (c) and 2$t(0)=lim& a 0 2t (&)�&.Below, we prove that the last limit exists, and 2$0(0)=(m(c), h=).Inequality (B2) then follows from concavity of V(c) in c and the proof iscompleted as in Case 2.

Given any &>0, by the mean value theorem, we have

2t (&)&

=Et _|T

tfc(cs+`&

s , Vs(c+&h)) h=s+ fv(cs , Vs(c)+!&

s)2s(&)

&ds& ,


where `&s # [0, &hs], and !&

s # [0, 2s(&)] (with the convention [0, a]=[a, 0] if a<0). Since fv+(1+:) ;<0, the above linear SDE has theunique solution

2t (&)&

=Et _|T

tG&

t, s ds& ,

where G&t, s=exp(�s

t fv(cu , Vu(c)+!&u) du) fc(cs+`&

s , Vs(c+&h)) h=s .

To accommodate taking a limit, we derive uniform integrable bounds onG& for small &. Monotonicity of V(c) in c and fcc , fcv<0 imply that thereexists small enough &� >0, such that for all & # (0, &� ),

0� fc(cs+`&s , Vs(c+&h=))� fc(cs �2, Vs(c�2)). (B3)

Because the right-hand side of (B3) is proportional to fc(cs , Vs(c)), we haveuniform integrable bounds on G& if E �T&=

0 f 1+}c (cs , Vs(c)) ds<� for some

}>0 and every = # (0, T ). This condition is implied by the squareintegrability of m, the first-order conditions (1), our assumed restriction onfv , and an application of the Ho� lder inequality. A first implication of thesebounds is that lim& a 0 2t(&)=0, and therefore

lim& a 0

G&t, s=G 0

t, s #exp \|s

tfv(cu , Vu(c)) du+ fc(s, cs , Vs(c)) h=

s .

A second implication of the bounds on G& is that we can apply thedominated convergence theorem to conclude that 2$t(0)=Et[�T

t G0t, s ds],

completing the proof.

Case 4 (#<0 and :>0). In this case, we prove C2$ under therestriction E[exp(2 �T&=

0 fv(cs , Vs(c))) ds]<� for every = # (0, T ).

Let V� } denote the ordinally equivalent utility process as in (10) but withterminal value V� T=}, where } # R& . Also, let c}

t =max(ct , |}| ). ThenV� t(c)�e&;(T&t)} and fv(c}

t , V }t (c)) is uniformly bounded for any c # C. We

define 2}t (&)=V }

t (c+&h=)&V }t (c). As in Case 3, we use (B3), V} # D0 , and

the dominated convergence theorem to show that lim& a 0 2}0(&)�&=

(m}(c}), h=) where m}(c}) is given by (3) with V } taking the place of V.Concavity implies

V }0(c}+h=)�V }

0(c})+(m}(c), h=) , }<0.

We take the limit as } A 0 to obtain (B2) by using the dominatedconvergence theorem together with the assumed restriction on fv , and theuniform upper bound

m}t (c})�exp \|

t

0fv(cs , V 0

s(c)) ds+ fc(ct , V }�t (c)), }<}� .


The proof is completed as in Case 2 using the uniform (in =) bounds

f (ct �2, V(c�2))� f (ct+h=t , V(c+h=))�0.

Proof of Theorem 2

We prove Theorem 2 using the results of Theorems 3 and 4.

Part (a). Under condition I or III, the optimal consumption dynamicsare corollaries of Theorems 3 and 4. (For #=0, we obtain Z=0, and for#{0, we obtain :Z=0.)

Part (b). Using the notation of Eq. (17), we have the general identity

?scs

?tct=exp \|

s

t+c

{&r{&_c{ } '{ d{+ `s

`t,

where ` is the exponential supermartingale

`t=exp \|t

0& 1

2 &_c{&'${&2 d{+|

t

0_c

{&'${ dB{+ .

Suppose first that r and ' are deterministic. Then, by part (a) we have

+ct &rt&_c

t } 't=&qt , and _ct &'$t =(kt&1) '$t .

Since the appropriate Novikov condition is satisfied, it follows that ` is amartingale, and therefore

Wt(c)ct

=Et _|T

t

?scs

?t ctds&=|

T

texp \&|

s

tq{ d{+ ds.

Next, we consider the case of #=0, while r and ' are potentiallystochastic. By Theorem 3, we again have +c&r&_c } '=&q. Given theboundedness of r and ', ` is a martingale, and the same argument applies.A direct argument that does not rely on Theorem 3, beyond the fact thatthe unique solution c satisfies the first-order conditions, is as follows. Usingthe transformation of Lemma A1, we know that the utility processV=V(c) satisfies

1+:Vt=Et _exp \|T

t: log(cs)&; log(1+:Vs) ds+&

=e;(T&t)Et _exp \|T

tfv(cs , Vs) ds+& .


From this equation and (3), it follows that mt(c)=e;(T&t)Mt �ct , where Mis the martingale defined by

Mt=Et _exp \|T

0fv(cs , Vs) ds+& .

The first-order condition mt(c)=*?t can be restated as ct*?t=e;(T&t)Mt .Using the definition of the wealth process, we obtain Wt(c) *?t=;&1(e;(T&t)&1) Mt . Dividing the last two equations proves the result.

Part (c). The optimal trading strategy is computed as explained inSection 2.3. Under condition II or condition III, part (b) implies that_W(c)=_c. Under condition I or condition III, part (a) implies that_c=k'$. Combining these observations gives the result.

Proof of Lemma 3

Letting V and c be defined in terms of X and J by (14) and (15), respec-tively, it suffices to prove that Vt+� t

0 f (cs , Vs) ds is a martingale. FromEqs. (14) and (15) and the definition of f, we have

:f (ct , Vt)1+:Vt

= fv(ct , Vt)+;=((;&:) kt&;) Xt+(:&;) Jt .

Using these equations and (4a) in the Ito expansion of (14), we obtain

:1+:Vt

(dVt+ f (ct , Vt) dt)=[AtXt+Bt] dt+(Zt&(1&kt) '$t ) dBt ,

(B4)

where At=(;&:) k2t &;kt&k4 t , and, with +J denoting the drift term of J,

Bt=+Jt &(1&kt) \rt&;+

kt

2't } 't++kt(:&;) Jt

+12

Zt } Zt&(1&kt) Zt } 't .

(Hint: In computing A and B, express everything in terms of fv first, andsubstitute in the expression for fv last.) We are to show that both A andB vanish. In fact, the condition At=0, together with kT=1, is equivalentto (11) (with #=0), while the condition Bt=0, together with JT=0, isequivalent to (13). This completes the proof of the lemma.


Proof of Theorem 3

We have already argued that the unique solution to the first-order condi-tions is given by (15) for properly selected *. (That this solution is anelement of C follows by combining Lemma C1, (15), and the assumptionthat r and ' are bounded.) Applying Ito's lemma, we obtain

dct

ct=

de&Xt

e&Xt+

: dVt

1+:Vt+\de&Xt

e&Xt +\ : dVt

1+:Vt+ .

From the dynamics of X in (4a) and Lemma 3 (see Eq. (B4)), we have

de&Xt

e&Xt=( fv(ct , Vt)+rt+'t } 't) dt+'$ dBt ,

: dVt

1+:Vt=&

:f (ct , Vt)1+:Vt

dt+(Zt&(1&kt) '$t ) dBt .

Substituting back into the consumption dynamics and using the identity

:f (ct , Vt)1+:Vt

= fv(ct , Vt)+;,

the optimal consumption dynamics of Theorem 2 follow immediately.Optimality verification follows easily from Lemma 2, while uniqueness is

a consequence of strict concavity of the utility function.We proved Eq. (16) as part of Theorem 1, while the optimal trading

strategy follows by the argument in Section 2.3, after observing that, by(16), _W(c)=_c.

Proof of Lemma 4

Letting V and c be defined in terms of X and J by (20) and (21), respec-tively, it suffices to prove that Vt+� t

0 f (cs , Vs) ds is a martingale. We define+~ J and _J so that

dJt

Jt=+~ J

t dt+_Jt dB� t=(+~ J

t +(1&k) _Jt } 't) dt+_J

t dBt .

We can now expand (20) using Ito's lemma, the dynamics of X in (4a), andthe identity

f (c, v)v

=1+:

:( fv(c, v)+;)


to obtain

dVt+ f (ct , Vt) dtVt

=(1+:k) _At dt+\_Jt +

#1&#

'$t+ dBt& , (B5)

where

At=+~ Jt +

fv(ct , Vt):

+#

1&# \rt+k2

't } 't++:k2

_Jt } _J

t +1

1&#;:k

.

We are therefore to show that At=0. To do so, we begin with the expression

fv(ct , Vt)=:c#

t

#V &1�(1+:)

t &(1+:) ;.

Using Eqs. (20) and (21) and simplifying, we observe that the coefficient ofXt in the exponent vanishes, resulting in

fv(ct , Vt):

=1#

(1+:)#�(1&#) J &1t &

1+::

;. (B6)

Given this expression, the backward SDE (18) (where Zt=_Jt Jt) is clearly

equivalent to At=0. Using the above expression for fv in (4a) results in theclaimed dynamics for X.

Proof of Theorem 4

Suppose that the assumptions of Theorem 4 are satisfied and that c isgiven by (21) (as constructed in Section 5.3). The following lemma showsthat c # C.

Lemma B1. The process |Jt|: exp(� t

0 fv(cs , Vs(c)) ds) is bounded aboveand away from zero (by deterministic constants, and uniformly in time).

Proof. Fix some t # [0, T), and let

js=#Js exp _|s

tau du& , s�t,

where au=#(1&#)&1 [ru&(;�#)+(k�2) 'u } 'u]. Then, with B� defined in(12), we have

djs=&_(1+:)#�(1&#) exp \|s

tau du++

:k2

zs } zs

js & dt+zs dB� s , jT=0.


By the boundedness of r and ', we choose a K>0 such that |at|<K forall t # [0, T]. Using the monotonicity result in Theorem A2, upper andlower bounds for jt are found by replacing au with K and &K, respectively,when u�t. This results in the following bounds for Jt :

1&e&K(T&t)

K�#Jt(1+:)&#�(1&#)�

eK(T&t)&1K

.

Using Eq. (B6) and integrating, it follows that, for some constantsC1 , C2>0 that do not depend on the choice of t,

C1(eK(T&t)&1)&:�exp \|t

0fv(s) ds+�C2(eK(T&t)&1)&:.

From the same bounds on #Jt we also get

C3(eK(T&t)&1):�|Jt|:�C4(eK(T&t)&1):,

for some constants C3 , C4>0 (independent of t). Multiplying the last twosets of inequalities completes the proof of Lemma B1. K

Returning to Eq. (21), and applying Ito's lemma, we obtain thedynamics:

dct

ct=&k dXt+

k2

2(dXt)

2+:kdJt

Jt+

:k(:k&1)2 \dJt

Jt +2

&:k2(dXt) \dJt

Jt + .

Simplifying the expression using (18) and the dynamics of X of Lemma 4results in the theorem's expressions for +c and _c. Optimality verificationfollows easily from Lemma 2 using

fv(ct , Vt)=:|#|

(1+:)#�(1&#) |Jt|&1&(1+:) ;.

The boundedness of r and ' and the bounds in Lemma A5 imply that forany =>0, |Jt| is uniformly bounded away from zero for t # [0, T&=].Uniqueness follows from strict concavity of the utility function.

To determine the optimal consumption and portfolio rules, we useLemma B2 below in conjunction with Eq. (28) to obtain

Wt(c)=(1+:) #Vt exp(&Xt). (B7)


Equations (B7), (20), and (21) combined give the claimed optimalconsumption-to-wealth ratio. Using expression (B5) (where _J

t =Zt�J t) and(B7), we find

_W(c)t =k'$+(1+:k)

Zt

Jt.

Arguing as in Section 2.3 we obtain the claimed optimal portfolioallocation.

Lemma B2. For all t<T,

?t Wt(c)=*&1(1+:) #Vt exp \|t

0fv(s) ds+ , #{0.

Proof. We start with the Ito expansion

Et _VT&= exp \|T&=

0fv(s) ds+&&Vt exp \|

t

0fv(s) ds+

=Et _|T&=

texp \|

s

0fv(u) du+ (Vs fv(s)& f (s)) ds& ,

for any = # (0, T ). In eliminating the martingale part, we have used the factthat E exp(�T&=

0 2fv(u) du)<� (which follows from fv�0 if :�0 and fromthe boundedness of r and ' if :>0) and the fact that E(�T&=

0 Z$sZs ds)<�for every = # (0, T) (which follows from the Burkholder�Davis�Gundyinequality and the bounds in Lemma A5). Substituting #[Vs fv(s)& f (s)]=&c#

s |Vs|:�(1+:) and using the first-order condition

(1+:) exp \|s

0fv(u) du+ c#

s |Vs|:�(1+:)=*cs?s

we obtain, for all = # (0, T ),

*&1(1+:) #Vt exp \|t

0fv(s) ds+

=Et _|T&=

t?scs ds&+*&1(1+:) #Et _VT&= exp \|

T&=

0fv(s) ds+& .

The proof is completed by letting = approach zero, provided that we show

limt A T

Es _Vt exp \|t

0fv(s) ds)&=0, s # [0, T). (B8)


If :�0, then fv�0 and (B8) follows trivially since limt A T Es(Vt)=0. If:>0, we use (20) and the definition of X to get

Vt exp \|t

0fv(s) ds+=(#�|#| ) *1&k?1&k

t |Jt|1+:k exp \|

t

0kfv(s) ds+ .

The limit (B8) then follows by Lemma B1. K

Proof of Lemma 5

We first show that H�0. We have the upper bound for J from (13)

Jt�K&Et _|T

texp \|

s

t(:&;) ku du+ (1&ks) \b+

ks

2(82)$ `s+ Ys ds&

for some constant K. If :<0 then kt<1 for t<T. The restrictions on !along with Yt # Y imply that Y has a uniform constant lower bound, whichimplies a constant upper bound on J. This is contradicted, however, ifH i

t>0 for some i since we can consider starting the Y process at time twith Y i

t arbitrarily large, violating the upper bound on J. It remains toshow that H has a finite lower bound. The ODE for H is of the form

H4 {=P{+Q{H{+ 12`$(7$Ht)

2, H0=0,

where P{ # Rn, Q{ # Rn_n, and {=T&t measures time from the horizondate. For any K # R,

1d{

1$e&K{H{=1$e&K{[P{+(Q{&KI ) H{+12

`$(7$Ht)2]

�1$e&K{[P{+(Q{&KI ) H{]

Because Q is uniformly bounded, we can choose K large enough so that1$(Q{&KI )�0, for all {�0, which, combined with H�0 implies1$H{�1$ �{

0 eK({&s)Ps ds.

Proof of Lemma 6

The ODE for F is of the form

F4 {=P{+Q${Ft+FtQ{+2F{ 77$F{ , F0=0,

where P{ # Rn, Q{ # Rn_n, and {=T&t measures time from the horizondate. The matrix P is negative semidefinite. Lemma 6 then follows from the


following claim, which can be shown by adapting the proof of Lemma16.4.2 in Lancaster and Rodman [36] to the variable coefficient case: Forany z # Rn,

z$F{z= maxh bounded {|

{

0Z$s (P{&s&2h$shs) Zs ds,

where Z is Rn valued and satisfies Z4 s=(Q{&s&27hs) Zs , Z0=z.

APPENDIX C

Extensions of the Gronwall�Bellman Inequality

In this appendix we derive some generalizations of the ``stochasticGronwall�Bellman'' inequality of Duffie and Epstein [12], results that areused in several of our proofs. The probabilistic setting and notation arethe same as in Appendix A. We also use standard lattice notation:x7 y=min[x, y], x 6 y=max[x, y], x+=max[x, 0].

Lemma C1. Suppose that : # Dexp1 , ; # D1 , x # D0 , and = # R. If :t�0

and xt�Et[�Tt (:sxs+;s) ds+=] for all t # [0, T], then

xt�Et _|T

t;s exp \|

s

t:u du+ ds+= exp \|

T

t:u du+& , t # [0, T].

The result is also valid with the last two inequalities reversed.

Proof. Let yt represent the right-hand side of the last inequality. It caneasily be shown that yt=Et[�T

t (:s ys+;s) ds+=] for all t. Letting$=x& y, it follows that $t�Et[�T

t :s$s ds] for all t. Let now At=exp(� t

0 :s ds), Ct=� t0 :s$s ds, and Mt=Et(CT). We observe that M is a

martingale, and $�M&C. It follows that d(At Ct)=:tAt(Ct+$t) dt�Mt dAt=d(AtMt)&At dMt . Integrating from t to T and applying theoperator Et gives M&C�0, and therefore $�0. The above argument isalso valid with all inequalities reversed. K

Lemma C2. Suppose that :+ # Dexp1 , ; # D1 , x # D0 , and x is right-

continuous. If for every stopping time {, xt�Et[�{t (:sxs+;s) ds+x{] on

[t<{], t # [0, T], and xT�0, then

xt�Et _|T

t;s exp \|

s

t:u du+ ds& , t # [0, T].

The result is also valid with the last three inequalities reversed.


Proof. The same argument as in the first part of Lemma C1 shows thatit is enough to prove the result when ; vanishes. Assuming ;=0, supposethat, for some time t, the event A=[xt>0] is non-null, and define{=inf[s�t : xs�0]. Since x is assumed right-continuous, x{�0, andxs>0 on A & [t�s<{]. It follows that, on the event A & [t�s<{],xs�Es[�{

s :u xu du]. Therefore,

xs�Et _|T

s:+

u 1[u�{] xu ds& on A, s # [t, T].

Applying Lemma C1 on A_[t, T], we conclude that xt�0 on A,a contradiction. K

A more elaborate argument in the following lemma shows that, for ;=0,Lemma C2 is valid under much weaker integrability assumptions.

Lemma C3. Suppose that : and x are progressively measurable processessatisfying E[�T

0 :+t dt]<� and E[�T

0 :+t x+

t dt]<�. Suppose further thatx is right-continuous, and, for every stopping time {,

xt�Et _|{

t:s xs ds+x{& on [{>t], (C1)

and xT�0. Then xt�0 for all t.

Proof. Suppose, to the contrary, that, for some time t, the eventA=[xt>0] is of positive probability. In the argument that followswe restrict time to the interval [t, T]. Consider the stopping time{=inf[s : s�t, xs�0]. Since x is right-continuous, x{�0, and xs>0 onA & [t�s<{]. Moreover, defining the process Xs=Es[�{ 6 s

s :+u xu du],

s # [t, T], we have, from (C1), that

xs�Xs on [s�{], for all s�t, (C2)

and in particular, Xt>0 on A, while of course X{=0. We also define thestopping times

{n=inf {s : s�t, Xs�1n= 1A+t10"A , n=1, 2, ...,


and the martingales

Ms=Es _|{

t:+

u xu du& , N ns =|

s 7{n

t

dMu

Xu, s�t, n=1, 2, ... .

By Ito's lemma,

log \X{n

Xt +=|{n

t

dXs

Xs&

1

2 \dXs

Xs +2

�|{n

t

dXs

Xs

=&|{n

t

:+s xs

Xsds+N n

{ &N nt .

Since Xt>0 on A, there exists =>0 small enough so that the event[Xt>=] is of positive probability. Fixing such an =, and using the fact thatX{n

�1�n, we obtain

log(=n)�E _|{n

t

:+s xs

Xsds&�E _|

T

0:+

s ds& ,

where the last inequality follows from (C2). Letting n approach infinity, wereach the conclusion that E[�T

0 :+s ds]=�, contradicting the lemma's

assumptions. K

The above lemma is not valid without the assumption E[�T0 :+

t dt]<�.For example, if :t=(T&t)&1, then (C1) is satisfied as an equality byxt=Mt(T&t) for any martingale M.

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